Simplify ((x^2-64)^5)/((x^2-16x+64) square root of x+8)

$\frac{{(x^{2} - 64)}^{5}}{(x^{2} - 16 x + 64) \sqrt{x + 8}}$

Step 1

Simplify the numerator.

Rewrite $64$ as $8^{2}$ .

$\frac{{(x^{2} - 8^{2})}^{5}}{(x^{2} - 16 x + 64) \sqrt{x + 8}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$\frac{{((x + 8) (x - 8))}^{5}}{(x^{2} - 16 x + 64) \sqrt{x + 8}}$

Apply the product rule to $(x + 8) (x - 8)$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{5}}{(x^{2} - 16 x + 64) \sqrt{x + 8}}$

Step 2

Factor using the perfect square rule.

Rewrite $64$ as $8^{2}$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{5}}{(x^{2} - 16 x + 8^{2}) \sqrt{x + 8}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$\frac{{(x + 8)}^{5} {(x - 8)}^{5}}{(x^{2} - 2 \cdot x \cdot 8 + 8^{2}) \sqrt{x + 8}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 8$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{5}}{{(x - 8)}^{2} \sqrt{x + 8}}$

Step 3

Cancel the common factor of ${(x - 8)}^{5}$ and ${(x - 8)}^{2}$ .

Factor ${(x - 8)}^{2}$ out of ${(x + 8)}^{5} {(x - 8)}^{5}$ .

$\frac{{(x - 8)}^{2} ({(x + 8)}^{5} {(x - 8)}^{3})}{{(x - 8)}^{2} \sqrt{x + 8}}$

Cancel the common factors.

Factor ${(x - 8)}^{2}$ out of ${(x - 8)}^{2} \sqrt{x + 8}$ .

$\frac{{(x - 8)}^{2} ({(x + 8)}^{5} {(x - 8)}^{3})}{{(x - 8)}^{2} (\sqrt{x + 8})}$

Cancel the common factor.

$\frac{{(x - 8)}^{2} ({(x + 8)}^{5} {(x - 8)}^{3})}{{(x - 8)}^{2} \sqrt{x + 8}}$

Rewrite the expression.

$\frac{{(x + 8)}^{5} {(x - 8)}^{3}}{\sqrt{x + 8}}$

Step 4

Multiply $\frac{{(x + 8)}^{5} {(x - 8)}^{3}}{\sqrt{x + 8}}$ by $\frac{\sqrt{x + 8}}{\sqrt{x + 8}}$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3}}{\sqrt{x + 8}} \cdot \frac{\sqrt{x + 8}}{\sqrt{x + 8}}$

Step 5

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{{(x + 8)}^{5} {(x - 8)}^{3}}{\sqrt{x + 8}}$ by $\frac{\sqrt{x + 8}}{\sqrt{x + 8}}$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{\sqrt{x + 8} \sqrt{x + 8}}$

Raise $\sqrt{x + 8}$ to the power of $1$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{\sqrt{x + 8}}^{1} \sqrt{x + 8}}$

Raise $\sqrt{x + 8}$ to the power of $1$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{\sqrt{x + 8}}^{1} {\sqrt{x + 8}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{\sqrt{x + 8}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{\sqrt{x + 8}}^{2}}$

Rewrite ${\sqrt{x + 8}}^{2}$ as $x + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{({(x + 8)}^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{(x + 8)}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{(x + 8)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{(x + 8)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{{(x + 8)}^{1}}$

Simplify.

$\frac{{(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}}{x + 8}$

Cancel the common factor of ${(x + 8)}^{5}$ and $x + 8$ .

Factor $x + 8$ out of ${(x + 8)}^{5} {(x - 8)}^{3} \sqrt{x + 8}$ .

$\frac{(x + 8) ({(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8})}{x + 8}$

Cancel the common factors.

Multiply by $1$ .

$\frac{(x + 8) ({(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8})}{(x + 8) \cdot 1}$

Cancel the common factor.

$\frac{(x + 8) ({(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8})}{(x + 8) \cdot 1}$

Rewrite the expression.

$\frac{{(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8}}{1}$

Divide ${(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8}$ by $1$ .

${(x + 8)}^{4} {(x - 8)}^{3} \sqrt{x + 8}$

Step 6

Use the Binomial Theorem.

$(x^{4} + 4 x^{3} \cdot 8 + 6 x^{2} \cdot 8^{2} + 4 x \cdot 8^{3} + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Step 7

Simplify each term.

Multiply $8$ by $4$ .

$(x^{4} + 32 x^{3} + 6 x^{2} \cdot 8^{2} + 4 x \cdot 8^{3} + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Raise $8$ to the power of $2$ .

$(x^{4} + 32 x^{3} + 6 x^{2} \cdot 64 + 4 x \cdot 8^{3} + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Multiply $64$ by $6$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 4 x \cdot 8^{3} + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Raise $8$ to the power of $3$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 4 x \cdot 512 + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Multiply $512$ by $4$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 8^{4}) {(x - 8)}^{3} \sqrt{x + 8}$

Raise $8$ to the power of $4$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) {(x - 8)}^{3} \sqrt{x + 8}$

Step 8

Use the Binomial Theorem.

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} + 3 x^{2} \cdot - 8 + 3 x {(- 8)}^{2} + {(- 8)}^{3}) \sqrt{x + 8}$

Step 9

Simplify each term.

Multiply $- 8$ by $3$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} - 24 x^{2} + 3 x {(- 8)}^{2} + {(- 8)}^{3}) \sqrt{x + 8}$

Raise $- 8$ to the power of $2$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} - 24 x^{2} + 3 x \cdot 64 + {(- 8)}^{3}) \sqrt{x + 8}$

Multiply $64$ by $3$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} - 24 x^{2} + 192 x + {(- 8)}^{3}) \sqrt{x + 8}$

Raise $- 8$ to the power of $3$ .

$(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} - 24 x^{2} + 192 x - 512) \sqrt{x + 8}$

Step 10

Expand $(x^{4} + 32 x^{3} + 384 x^{2} + 2048 x + 4096) (x^{3} - 24 x^{2} + 192 x - 512)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{4} x^{3} + x^{4} (- 24 x^{2}) + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Step 11

Simplify terms.

Simplify each term.

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{4 + 3} + x^{4} (- 24 x^{2}) + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $4$ and $3$ .

$(x^{7} + x^{4} (- 24 x^{2}) + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{4} x^{2} + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 24 (x^{2} x^{4}) + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{2 + 4} + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $2$ and $4$ .

$(x^{7} - 24 x^{6} + x^{4} (192 x) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{4} x + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$(x^{7} - 24 x^{6} + 192 (x \cdot x^{4}) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 24 x^{6} + 192 (x^{1} x^{4}) + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{1 + 4} + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $1$ and $4$ .

$(x^{7} - 24 x^{6} + 192 x^{5} + x^{4} \cdot - 512 + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Move $- 512$ to the left of $x^{4}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 \cdot x^{4} + 32 x^{3} x^{3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 (x^{3} x^{3}) + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{3 + 3} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $3$ and $3$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} + 32 x^{3} (- 24 x^{2}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} + 32 \cdot - 24 x^{3} x^{2} + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} + 32 \cdot - 24 (x^{2} x^{3}) + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} + 32 \cdot - 24 x^{2 + 3} + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $2$ and $3$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} + 32 \cdot - 24 x^{5} + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $32$ by $- 24$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 x^{3} (192 x) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 \cdot 192 x^{3} x + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 \cdot 192 (x \cdot x^{3}) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 \cdot 192 (x^{1} x^{3}) + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 \cdot 192 x^{1 + 3} + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $1$ and $3$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 32 \cdot 192 x^{4} + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $32$ by $192$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} + 32 x^{3} \cdot - 512 + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $- 512$ by $32$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{2} x^{3} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 (x^{3} x^{2}) + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{3 + 2} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $3$ and $2$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} + 384 x^{2} (- 24 x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} + 384 \cdot - 24 x^{2} x^{2} + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} + 384 \cdot - 24 (x^{2} x^{2}) + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} + 384 \cdot - 24 x^{2 + 2} + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $2$ and $2$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} + 384 \cdot - 24 x^{4} + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $384$ by $- 24$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 x^{2} (192 x) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 \cdot 192 x^{2} x + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 \cdot 192 (x \cdot x^{2}) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 \cdot 192 (x^{1} x^{2}) + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 \cdot 192 x^{1 + 2} + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $1$ and $2$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 384 \cdot 192 x^{3} + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $384$ by $192$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} + 384 x^{2} \cdot - 512 + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $- 512$ by $384$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x \cdot x^{3} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 (x^{3} x) + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 (x^{3} x^{1}) + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{3 + 1} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $3$ and $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 x (- 24 x^{2}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 \cdot - 24 x \cdot x^{2} + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 \cdot - 24 (x^{2} x) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 \cdot - 24 (x^{2} x^{1}) + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 \cdot - 24 x^{2 + 1} + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Add $2$ and $1$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} + 2048 \cdot - 24 x^{3} + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $2048$ by $- 24$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 2048 x (192 x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Rewrite using the commutative property of multiplication.

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 2048 \cdot 192 x \cdot x + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 2048 \cdot 192 (x \cdot x) + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $x$ by $x$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 2048 \cdot 192 x^{2} + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $2048$ by $192$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} + 2048 x \cdot - 512 + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $- 512$ by $2048$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} + 4096 (- 24 x^{2}) + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $- 24$ by $4096$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 4096 (192 x) + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $192$ by $4096$ .

$(x^{7} - 24 x^{6} + 192 x^{5} - 512 x^{4} + 32 x^{6} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x + 4096 \cdot - 512) \sqrt{x + 8}$

Multiply $4096$ by $- 512$ .

Simplify terms.

Add $- 24 x^{6}$ and $32 x^{6}$ .

$(x^{7} + 8 x^{6} + 192 x^{5} - 512 x^{4} - 768 x^{5} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Subtract $768 x^{5}$ from $192 x^{5}$ .

$(x^{7} + 8 x^{6} - 576 x^{5} - 512 x^{4} + 6144 x^{4} - 16384 x^{3} + 384 x^{5} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 576 x^{5}$ and $384 x^{5}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 512 x^{4} + 6144 x^{4} - 16384 x^{3} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 512 x^{4}$ and $6144 x^{4}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} + 5632 x^{4} - 16384 x^{3} - 9216 x^{4} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Subtract $9216 x^{4}$ from $5632 x^{4}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 3584 x^{4} - 16384 x^{3} + 73728 x^{3} - 196608 x^{2} + 2048 x^{4} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 3584 x^{4}$ and $2048 x^{4}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} - 16384 x^{3} + 73728 x^{3} - 196608 x^{2} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 16384 x^{3}$ and $73728 x^{3}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 57344 x^{3} - 196608 x^{2} - 49152 x^{3} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Subtract $49152 x^{3}$ from $57344 x^{3}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 8192 x^{3} - 196608 x^{2} + 393216 x^{2} - 1048576 x + 4096 x^{3} - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $8192 x^{3}$ and $4096 x^{3}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 12288 x^{3} - 196608 x^{2} + 393216 x^{2} - 1048576 x - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 196608 x^{2}$ and $393216 x^{2}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 12288 x^{3} + 196608 x^{2} - 1048576 x - 98304 x^{2} + 786432 x - 2097152) \sqrt{x + 8}$

Subtract $98304 x^{2}$ from $196608 x^{2}$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 12288 x^{3} + 98304 x^{2} - 1048576 x + 786432 x - 2097152) \sqrt{x + 8}$

Add $- 1048576 x$ and $786432 x$ .

$(x^{7} + 8 x^{6} - 192 x^{5} - 1536 x^{4} + 12288 x^{3} + 98304 x^{2} - 262144 x - 2097152) \sqrt{x + 8}$

Apply the distributive property.

$x^{7} \sqrt{x + 8} + 8 x^{6} \sqrt{x + 8} - 192 x^{5} \sqrt{x + 8} - 1536 x^{4} \sqrt{x + 8} + 12288 x^{3} \sqrt{x + 8} + 98304 x^{2} \sqrt{x + 8} - 262144 x \sqrt{x + 8} - 2097152 \sqrt{x + 8}$

Simplify ((x^2-64)^5)/((x^2-16x+64) square root of x+8)

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