Simplify the Radical Expression ((8^4)/(10^4))^(-1/4)

${(\frac{8^{4}}{10^{4}})}^{- \frac{1}{4}}$

Step 1

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

${(\frac{4096}{10^{4}})}^{- \frac{1}{4}}$

Step 2

Raise $10$ to the power of $4$ .

${(\frac{4096}{10000})}^{- \frac{1}{4}}$

Step 3

Cancel the common factor of $4096$ and $10000$ .

Factor $16$ out of $4096$ .

${(\frac{16 (256)}{10000})}^{- \frac{1}{4}}$

Cancel the common factors.

Factor $16$ out of $10000$ .

${(\frac{16 \cdot 256}{16 \cdot 625})}^{- \frac{1}{4}}$

Cancel the common factor.

${(\frac{16 \cdot 256}{16 \cdot 625})}^{- \frac{1}{4}}$

Rewrite the expression.

${(\frac{256}{625})}^{- \frac{1}{4}}$

Step 4

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{625}{256})}^{\frac{1}{4}}$

Step 5

Apply the product rule to $\frac{625}{256}$ .

$\frac{625^{\frac{1}{4}}}{256^{\frac{1}{4}}}$

Step 6

Simplify the numerator.

Rewrite $625$ as $5^{4}$ .

$\frac{{(5^{4})}^{\frac{1}{4}}}{256^{\frac{1}{4}}}$

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

$\frac{5^{4 (\frac{1}{4})}}{256^{\frac{1}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{5^{4 (\frac{1}{4})}}{256^{\frac{1}{4}}}$

Rewrite the expression.

$\frac{5^{1}}{256^{\frac{1}{4}}}$

Evaluate the exponent.

$\frac{5}{256^{\frac{1}{4}}}$

Step 7

Simplify the denominator.

Rewrite $256$ as $4^{4}$ .

$\frac{5}{{(4^{4})}^{\frac{1}{4}}}$

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

$\frac{5}{4^{4 (\frac{1}{4})}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{5}{4^{4 (\frac{1}{4})}}$

Rewrite the expression.

$\frac{5}{4^{1}}$

Evaluate the exponent.

$\frac{5}{4}$

Simplify the Radical Expression ((8^4)/(10^4))^(-1/4)

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