Simplify the Radical Expression (32x^30y^-40)^(-2/5)

${(32 x^{30} y^{- 40})}^{- \frac{2}{5}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(32 x^{30} \frac{1}{y^{40}})}^{- \frac{2}{5}}$

Step 2

Multiply $32 x^{30} \frac{1}{y^{40}}$ .

Combine $32$ and $\frac{1}{y^{40}}$ .

${(x^{30} \frac{32}{y^{40}})}^{- \frac{2}{5}}$

Combine $x^{30}$ and $\frac{32}{y^{40}}$ .

${(\frac{x^{30} \cdot 32}{y^{40}})}^{- \frac{2}{5}}$

Step 3

Move $32$ to the left of $x^{30}$ .

${(\frac{32 \cdot x^{30}}{y^{40}})}^{- \frac{2}{5}}$

Step 4

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

${(\frac{y^{40}}{32 x^{30}})}^{\frac{2}{5}}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{y^{40}}{32 x^{30}}$ .

$\frac{{(y^{40})}^{\frac{2}{5}}}{{(32 x^{30})}^{\frac{2}{5}}}$

Apply the product rule to $32 x^{30}$ .

$\frac{{(y^{40})}^{\frac{2}{5}}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Step 6

Multiply the exponents in ${(y^{40})}^{\frac{2}{5}}$ .

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

$\frac{y^{40 (\frac{2}{5})}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Cancel the common factor of $5$ .

Factor $5$ out of $40$ .

$\frac{y^{5 (8) \frac{2}{5}}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Cancel the common factor.

$\frac{y^{5 \cdot 8 \frac{2}{5}}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Rewrite the expression.

$\frac{y^{8 \cdot 2}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Multiply $8$ by $2$ .

$\frac{y^{16}}{32^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

Step 7

Simplify the denominator.

Rewrite $32$ as $2^{5}$ .

$\frac{y^{16}}{{(2^{5})}^{\frac{2}{5}} {(x^{30})}^{\frac{2}{5}}}$

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

$\frac{y^{16}}{2^{5 (\frac{2}{5})} {(x^{30})}^{\frac{2}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{y^{16}}{2^{5 (\frac{2}{5})} {(x^{30})}^{\frac{2}{5}}}$

Rewrite the expression.

$\frac{y^{16}}{2^{2} {(x^{30})}^{\frac{2}{5}}}$

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

$\frac{y^{16}}{4 {(x^{30})}^{\frac{2}{5}}}$

Multiply the exponents in ${(x^{30})}^{\frac{2}{5}}$ .

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

$\frac{y^{16}}{4 x^{30 (\frac{2}{5})}}$

Cancel the common factor of $5$ .

Factor $5$ out of $30$ .

$\frac{y^{16}}{4 x^{5 (6) \frac{2}{5}}}$

Cancel the common factor.

$\frac{y^{16}}{4 x^{5 \cdot 6 \frac{2}{5}}}$

Rewrite the expression.

$\frac{y^{16}}{4 x^{6 \cdot 2}}$

Multiply $6$ by $2$ .

$\frac{y^{16}}{4 x^{12}}$

Simplify the Radical Expression (32x^30y^-40)^(-2/5)

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