Simplify - square root of (x^2-6)^8

$- \sqrt{{(x^{2} - 6)}^{8}}$

Rewrite ${(x^{2} - 6)}^{8}$ as ${({(x^{2} - 6)}^{4})}^{2}$ .

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

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

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

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Multiply the exponents in ${(x^{2})}^{4}$ .

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

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

Multiply $2$ by $4$ .

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

Multiply the exponents in ${(x^{2})}^{3}$ .

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

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

Multiply $2$ by $3$ .

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

Multiply $- 6$ by $4$ .

$- (x^{8} - 24 x^{6} + 6 {(x^{2})}^{2} {(- 6)}^{2} + 4 x^{2} {(- 6)}^{3} + {(- 6)}^{4})$

Multiply the exponents in ${(x^{2})}^{2}$ .

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

$- (x^{8} - 24 x^{6} + 6 x^{2 \cdot 2} {(- 6)}^{2} + 4 x^{2} {(- 6)}^{3} + {(- 6)}^{4})$

Multiply $2$ by $2$ .

$- (x^{8} - 24 x^{6} + 6 x^{4} {(- 6)}^{2} + 4 x^{2} {(- 6)}^{3} + {(- 6)}^{4})$

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

$- (x^{8} - 24 x^{6} + 6 x^{4} \cdot 36 + 4 x^{2} {(- 6)}^{3} + {(- 6)}^{4})$

Multiply $36$ by $6$ .

$- (x^{8} - 24 x^{6} + 216 x^{4} + 4 x^{2} {(- 6)}^{3} + {(- 6)}^{4})$

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

$- (x^{8} - 24 x^{6} + 216 x^{4} + 4 x^{2} \cdot - 216 + {(- 6)}^{4})$

Multiply $- 216$ by $4$ .

$- (x^{8} - 24 x^{6} + 216 x^{4} - 864 x^{2} + {(- 6)}^{4})$

Raise $- 6$ to the power of $4$ .

$- (x^{8} - 24 x^{6} + 216 x^{4} - 864 x^{2} + 1296)$

Apply the distributive property.

$- x^{8} - (- 24 x^{6}) - (216 x^{4}) - (- 864 x^{2}) - 1 \cdot 1296$

Simplify.

Multiply $- 24$ by $- 1$ .

$- x^{8} + 24 x^{6} - (216 x^{4}) - (- 864 x^{2}) - 1 \cdot 1296$

Multiply $216$ by $- 1$ .

$- x^{8} + 24 x^{6} - 216 x^{4} - (- 864 x^{2}) - 1 \cdot 1296$

Multiply $- 864$ by $- 1$ .

$- x^{8} + 24 x^{6} - 216 x^{4} + 864 x^{2} - 1 \cdot 1296$

Multiply $- 1$ by $1296$ .

$- x^{8} + 24 x^{6} - 216 x^{4} + 864 x^{2} - 1296$

Simplify - square root of (x^2-6)^8

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