Simplify fourth root of 81(x^4-16)^4

$\sqrt[4]{81 {(x^{4} - 16)}^{4}}$

Step 1

Simplify the expression.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\sqrt[4]{81 {({(x^{2})}^{2} - 16)}^{4}}$

Rewrite $16$ as $4^{2}$ .

$\sqrt[4]{81 {({(x^{2})}^{2} - 4^{2})}^{4}}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 4$ .

$\sqrt[4]{81 {((x^{2} + 4) (x^{2} - 4))}^{4}}$

Step 3

Simplify.

Rewrite $4$ as $2^{2}$ .

$\sqrt[4]{81 {((x^{2} + 4) (x^{2} - 2^{2}))}^{4}}$

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 $b = 2$ .

$\sqrt[4]{81 {((x^{2} + 4) (x + 2) (x - 2))}^{4}}$

Step 4

Apply the product rule to $(x^{2} + 4) (x + 2) (x - 2)$ .

$\sqrt[4]{81 {((x^{2} + 4) (x + 2))}^{4} {(x - 2)}^{4}}$

Step 5

Expand $(x^{2} + 4) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$\sqrt[4]{81 {(x^{2} (x + 2) + 4 (x + 2))}^{4} {(x - 2)}^{4}}$

Apply the distributive property.

$\sqrt[4]{81 {(x^{2} x + x^{2} \cdot 2 + 4 (x + 2))}^{4} {(x - 2)}^{4}}$

Apply the distributive property.

$\sqrt[4]{81 {(x^{2} x + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

Step 6

Simplify each term.

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

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

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

$\sqrt[4]{81 {(x^{2} x^{1} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

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

$\sqrt[4]{81 {(x^{2 + 1} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

Add $2$ and $1$ .

$\sqrt[4]{81 {(x^{3} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

Move $2$ to the left of $x^{2}$ .

$\sqrt[4]{81 {(x^{3} + 2 \cdot x^{2} + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

Multiply $4$ by $2$ .

$\sqrt[4]{81 {(x^{3} + 2 x^{2} + 4 x + 8)}^{4} {(x - 2)}^{4}}$

Step 7

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\sqrt[4]{81 {((x^{3} + 2 x^{2}) + 4 x + 8)}^{4} {(x - 2)}^{4}}$

Factor out the greatest common factor (GCF) from each group.

$\sqrt[4]{81 {(x^{2} (x + 2) + 4 (x + 2))}^{4} {(x - 2)}^{4}}$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$\sqrt[4]{81 {((x + 2) (x^{2} + 4))}^{4} {(x - 2)}^{4}}$

Step 9

Apply the product rule to $(x + 2) (x^{2} + 4)$ .

$\sqrt[4]{81 {(x + 2)}^{4} {(x^{2} + 4)}^{4} {(x - 2)}^{4}}$

Step 10

Rewrite $81 {(x + 2)}^{4} {(x^{2} + 4)}^{4} {(x - 2)}^{4}$ as ${(3 (x + 2) (x^{2} + 4) (x - 2))}^{4}$ .

$\sqrt[4]{{(3 (x + 2) (x^{2} + 4) (x - 2))}^{4}}$

Step 11

Pull terms out from under the radical, assuming positive real numbers.

$3 (x + 2) (x^{2} + 4) (x - 2)$

Step 12

Simplify by multiplying through.

Apply the distributive property.

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

Multiply $3$ by $2$ .

$(3 x + 6) (x^{2} + 4) (x - 2)$

Step 13

Expand $(3 x + 6) (x^{2} + 4)$ using the FOIL Method.

Apply the distributive property.

$(3 x (x^{2} + 4) + 6 (x^{2} + 4)) (x - 2)$

Apply the distributive property.

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

Apply the distributive property.

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

Step 14

Simplify each term.

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

Move $x^{2}$ .

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

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

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

$(3 (x^{2} x^{1}) + 3 x \cdot 4 + 6 x^{2} + 6 \cdot 4) (x - 2)$

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

$(3 x^{2 + 1} + 3 x \cdot 4 + 6 x^{2} + 6 \cdot 4) (x - 2)$

Add $2$ and $1$ .

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

Multiply $4$ by $3$ .

$(3 x^{3} + 12 x + 6 x^{2} + 6 \cdot 4) (x - 2)$

Multiply $6$ by $4$ .

$(3 x^{3} + 12 x + 6 x^{2} + 24) (x - 2)$

Step 15

Expand $(3 x^{3} + 12 x + 6 x^{2} + 24) (x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{3} x + 3 x^{3} \cdot - 2 + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Step 16

Simplify terms.

Simplify each term.

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

Move $x$ .

$3 (x \cdot x^{3}) + 3 x^{3} \cdot - 2 + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

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

$3 (x^{1} x^{3}) + 3 x^{3} \cdot - 2 + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

$3 x^{1 + 3} + 3 x^{3} \cdot - 2 + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Add $1$ and $3$ .

$3 x^{4} + 3 x^{3} \cdot - 2 + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Multiply $- 2$ by $3$ .

$3 x^{4} - 6 x^{3} + 12 x \cdot x + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

Move $x$ .

$3 x^{4} - 6 x^{3} + 12 (x \cdot x) + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Multiply $x$ by $x$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} + 12 x \cdot - 2 + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Multiply $- 2$ by $12$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{2} x + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

Move $x$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 (x \cdot x^{2}) + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

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

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 (x^{1} x^{2}) + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

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

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{1 + 2} + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Add $1$ and $2$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{3} + 6 x^{2} \cdot - 2 + 24 x + 24 \cdot - 2$

Multiply $- 2$ by $6$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{3} - 12 x^{2} + 24 x + 24 \cdot - 2$

Multiply $24$ by $- 2$ .

$3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{3} - 12 x^{2} + 24 x - 48$

Combine the opposite terms in $3 x^{4} - 6 x^{3} + 12 x^{2} - 24 x + 6 x^{3} - 12 x^{2} + 24 x - 48$ .

Add $- 6 x^{3}$ and $6 x^{3}$ .

$3 x^{4} + 12 x^{2} - 24 x + 0 - 12 x^{2} + 24 x - 48$

Add $3 x^{4} + 12 x^{2} - 24 x$ and $0$ .

$3 x^{4} + 12 x^{2} - 24 x - 12 x^{2} + 24 x - 48$

Subtract $12 x^{2}$ from $12 x^{2}$ .

$3 x^{4} - 24 x + 0 + 24 x - 48$

Add $3 x^{4} - 24 x$ and $0$ .

$3 x^{4} - 24 x + 24 x - 48$

Add $- 24 x$ and $24 x$ .

$3 x^{4} + 0 - 48$

Add $3 x^{4}$ and $0$ .

$3 x^{4} - 48$

Simplify fourth root of 81(x^4-16)^4

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