Simplify ((x^2-36)^4)/((x^2-12x+36) square root of x+6)

$\frac{{(x^{2} - 36)}^{4}}{(x^{2} - 12 x + 36) \sqrt{x + 6}}$

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{{(x^{2} - 6^{2})}^{4}}{(x^{2} - 12 x + 36) \sqrt{x + 6}}$

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 $b = 6$ .

$\frac{{((x + 6) (x - 6))}^{4}}{(x^{2} - 12 x + 36) \sqrt{x + 6}}$

Apply the product rule to $(x + 6) (x - 6)$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{4}}{(x^{2} - 12 x + 36) \sqrt{x + 6}}$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{4}}{(x^{2} - 12 x + 6^{2}) \sqrt{x + 6}}$

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

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{4}}{{(x - 6)}^{2} \sqrt{x + 6}}$

Cancel the common factor of ${(x - 6)}^{4}$ and ${(x - 6)}^{2}$ .

Factor ${(x - 6)}^{2}$ out of ${(x + 6)}^{4} {(x - 6)}^{4}$ .

$\frac{{(x - 6)}^{2} ({(x + 6)}^{4} {(x - 6)}^{2})}{{(x - 6)}^{2} \sqrt{x + 6}}$

Cancel the common factors.

Factor ${(x - 6)}^{2}$ out of ${(x - 6)}^{2} \sqrt{x + 6}$ .

$\frac{{(x - 6)}^{2} ({(x + 6)}^{4} {(x - 6)}^{2})}{{(x - 6)}^{2} (\sqrt{x + 6})}$

Cancel the common factor.

$\frac{{(x - 6)}^{2} ({(x + 6)}^{4} {(x - 6)}^{2})}{{(x - 6)}^{2} \sqrt{x + 6}}$

Rewrite the expression.

$\frac{{(x + 6)}^{4} {(x - 6)}^{2}}{\sqrt{x + 6}}$

Multiply $\frac{{(x + 6)}^{4} {(x - 6)}^{2}}{\sqrt{x + 6}}$ by $\frac{\sqrt{x + 6}}{\sqrt{x + 6}}$ .

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

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{{(x + 6)}^{4} {(x - 6)}^{2}}{\sqrt{x + 6}}$ by $\frac{\sqrt{x + 6}}{\sqrt{x + 6}}$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{\sqrt{x + 6} \sqrt{x + 6}}$

Raise $\sqrt{x + 6}$ to the power of $1$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{\sqrt{x + 6}}^{1} \sqrt{x + 6}}$

Raise $\sqrt{x + 6}$ to the power of $1$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{\sqrt{x + 6}}^{1} {\sqrt{x + 6}}^{1}}$

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

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{\sqrt{x + 6}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{\sqrt{x + 6}}^{2}}$

Rewrite ${\sqrt{x + 6}}^{2}$ as $x + 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{({(x + 6)}^{\frac{1}{2}})}^{2}}$

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

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{(x + 6)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{(x + 6)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{(x + 6)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{{(x + 6)}^{1}}$

Simplify.

$\frac{{(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}}{x + 6}$

Cancel the common factor of ${(x + 6)}^{4}$ and $x + 6$ .

Factor $x + 6$ out of ${(x + 6)}^{4} {(x - 6)}^{2} \sqrt{x + 6}$ .

$\frac{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{2} \sqrt{x + 6})}{x + 6}$

Cancel the common factors.

Multiply by $1$ .

$\frac{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{2} \sqrt{x + 6})}{(x + 6) \cdot 1}$

Cancel the common factor.

$\frac{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{2} \sqrt{x + 6})}{(x + 6) \cdot 1}$

Rewrite the expression.

$\frac{{(x + 6)}^{3} {(x - 6)}^{2} \sqrt{x + 6}}{1}$

Divide ${(x + 6)}^{3} {(x - 6)}^{2} \sqrt{x + 6}$ by $1$ .

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

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Multiply $6$ by $3$ .

$(x^{3} + 18 x^{2} + 3 x \cdot 6^{2} + 6^{3}) {(x - 6)}^{2} \sqrt{x + 6}$

Raise $6$ to the power of $2$ .

$(x^{3} + 18 x^{2} + 3 x \cdot 36 + 6^{3}) {(x - 6)}^{2} \sqrt{x + 6}$

Multiply $36$ by $3$ .

$(x^{3} + 18 x^{2} + 108 x + 6^{3}) {(x - 6)}^{2} \sqrt{x + 6}$

Raise $6$ to the power of $3$ .

$(x^{3} + 18 x^{2} + 108 x + 216) {(x - 6)}^{2} \sqrt{x + 6}$

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$(x^{3} + 18 x^{2} + 108 x + 216) ((x - 6) (x - 6)) \sqrt{x + 6}$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$(x^{3} + 18 x^{2} + 108 x + 216) (x (x - 6) - 6 (x - 6)) \sqrt{x + 6}$

Apply the distributive property.

$(x^{3} + 18 x^{2} + 108 x + 216) (x \cdot x + x \cdot - 6 - 6 (x - 6)) \sqrt{x + 6}$

Apply the distributive property.

$(x^{3} + 18 x^{2} + 108 x + 216) (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) \sqrt{x + 6}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{3} + 18 x^{2} + 108 x + 216) (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) \sqrt{x + 6}$

Move $- 6$ to the left of $x$ .

$(x^{3} + 18 x^{2} + 108 x + 216) (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) \sqrt{x + 6}$

Multiply $- 6$ by $- 6$ .

$(x^{3} + 18 x^{2} + 108 x + 216) (x^{2} - 6 x - 6 x + 36) \sqrt{x + 6}$

Subtract $6 x$ from $- 6 x$ .

$(x^{3} + 18 x^{2} + 108 x + 216) (x^{2} - 12 x + 36) \sqrt{x + 6}$

Expand $(x^{3} + 18 x^{2} + 108 x + 216) (x^{2} - 12 x + 36)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{3} x^{2} + x^{3} (- 12 x) + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Simplify terms.

Simplify each term.

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

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

$(x^{3 + 2} + x^{3} (- 12 x) + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Add $3$ and $2$ .

$(x^{5} + x^{3} (- 12 x) + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{3} x + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

Move $x$ .

$(x^{5} - 12 (x \cdot x^{3}) + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

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

$(x^{5} - 12 (x^{1} x^{3}) + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

$(x^{5} - 12 x^{1 + 3} + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Add $1$ and $3$ .

$(x^{5} - 12 x^{4} + x^{3} \cdot 36 + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Move $36$ to the left of $x^{3}$ .

$(x^{5} - 12 x^{4} + 36 \cdot x^{3} + 18 x^{2} x^{2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

Move $x^{2}$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 (x^{2} x^{2}) + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{2 + 2} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Add $2$ and $2$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 x^{2} (- 12 x) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 \cdot - 12 x^{2} x + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

Move $x$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 \cdot - 12 (x \cdot x^{2}) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

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

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 \cdot - 12 (x^{1} x^{2}) + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 \cdot - 12 x^{1 + 2} + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Add $1$ and $2$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} + 18 \cdot - 12 x^{3} + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $18$ by $- 12$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 18 x^{2} \cdot 36 + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $36$ by $18$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x \cdot x^{2} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

Move $x^{2}$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 (x^{2} x) + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

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

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 (x^{2} x^{1}) + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{2 + 1} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Add $2$ and $1$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} + 108 x (- 12 x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} + 108 \cdot - 12 x \cdot x + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

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

Move $x$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} + 108 \cdot - 12 (x \cdot x) + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $x$ by $x$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} + 108 \cdot - 12 x^{2} + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $108$ by $- 12$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 108 x \cdot 36 + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $36$ by $108$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 3888 x + 216 x^{2} + 216 (- 12 x) + 216 \cdot 36) \sqrt{x + 6}$

Multiply $- 12$ by $216$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 3888 x + 216 x^{2} - 2592 x + 216 \cdot 36) \sqrt{x + 6}$

Multiply $216$ by $36$ .

$(x^{5} - 12 x^{4} + 36 x^{3} + 18 x^{4} - 216 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 3888 x + 216 x^{2} - 2592 x + 7776) \sqrt{x + 6}$

Simplify terms.

Add $- 12 x^{4}$ and $18 x^{4}$ .

$(x^{5} + 6 x^{4} + 36 x^{3} - 216 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 3888 x + 216 x^{2} - 2592 x + 7776) \sqrt{x + 6}$

Subtract $216 x^{3}$ from $36 x^{3}$ .

$(x^{5} + 6 x^{4} - 180 x^{3} + 648 x^{2} + 108 x^{3} - 1296 x^{2} + 3888 x + 216 x^{2} - 2592 x + 7776) \sqrt{x + 6}$

Add $- 180 x^{3}$ and $108 x^{3}$ .

$(x^{5} + 6 x^{4} - 72 x^{3} + 648 x^{2} - 1296 x^{2} + 3888 x + 216 x^{2} - 2592 x + 7776) \sqrt{x + 6}$

Subtract $1296 x^{2}$ from $648 x^{2}$ .

$(x^{5} + 6 x^{4} - 72 x^{3} - 648 x^{2} + 3888 x + 216 x^{2} - 2592 x + 7776) \sqrt{x + 6}$

Add $- 648 x^{2}$ and $216 x^{2}$ .

$(x^{5} + 6 x^{4} - 72 x^{3} - 432 x^{2} + 3888 x - 2592 x + 7776) \sqrt{x + 6}$

Subtract $2592 x$ from $3888 x$ .

$(x^{5} + 6 x^{4} - 72 x^{3} - 432 x^{2} + 1296 x + 7776) \sqrt{x + 6}$

Apply the distributive property.

$x^{5} \sqrt{x + 6} + 6 x^{4} \sqrt{x + 6} - 72 x^{3} \sqrt{x + 6} - 432 x^{2} \sqrt{x + 6} + 1296 x \sqrt{x + 6} + 7776 \sqrt{x + 6}$

Simplify ((x^2-36)^4)/((x^2-12x+36) square root of x+6)

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