Simplify the Radical Expression (64x^18y^-21)^(-4/3)

${(64 x^{18} y^{- 21})}^{- \frac{4}{3}}$

Step 1

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

${(64 x^{18} \frac{1}{y^{21}})}^{- \frac{4}{3}}$

Step 2

Multiply $64 x^{18} \frac{1}{y^{21}}$ .

Combine $64$ and $\frac{1}{y^{21}}$ .

${(x^{18} \frac{64}{y^{21}})}^{- \frac{4}{3}}$

Combine $x^{18}$ and $\frac{64}{y^{21}}$ .

${(\frac{x^{18} \cdot 64}{y^{21}})}^{- \frac{4}{3}}$

Step 3

Move $64$ to the left of $x^{18}$ .

${(\frac{64 \cdot x^{18}}{y^{21}})}^{- \frac{4}{3}}$

Step 4

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

${(\frac{y^{21}}{64 x^{18}})}^{\frac{4}{3}}$

Step 5

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

Apply the product rule to $\frac{y^{21}}{64 x^{18}}$ .

$\frac{{(y^{21})}^{\frac{4}{3}}}{{(64 x^{18})}^{\frac{4}{3}}}$

Apply the product rule to $64 x^{18}$ .

$\frac{{(y^{21})}^{\frac{4}{3}}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Step 6

Multiply the exponents in ${(y^{21})}^{\frac{4}{3}}$ .

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

$\frac{y^{21 (\frac{4}{3})}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Cancel the common factor of $3$ .

Factor $3$ out of $21$ .

$\frac{y^{3 (7) \frac{4}{3}}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Cancel the common factor.

$\frac{y^{3 \cdot 7 \frac{4}{3}}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Rewrite the expression.

$\frac{y^{7 \cdot 4}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Multiply $7$ by $4$ .

$\frac{y^{28}}{64^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

Step 7

Simplify the denominator.

Rewrite $64$ as $4^{3}$ .

$\frac{y^{28}}{{(4^{3})}^{\frac{4}{3}} {(x^{18})}^{\frac{4}{3}}}$

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

$\frac{y^{28}}{4^{3 (\frac{4}{3})} {(x^{18})}^{\frac{4}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{y^{28}}{4^{3 (\frac{4}{3})} {(x^{18})}^{\frac{4}{3}}}$

Rewrite the expression.

$\frac{y^{28}}{4^{4} {(x^{18})}^{\frac{4}{3}}}$

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

$\frac{y^{28}}{256 {(x^{18})}^{\frac{4}{3}}}$

Multiply the exponents in ${(x^{18})}^{\frac{4}{3}}$ .

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

$\frac{y^{28}}{256 x^{18 (\frac{4}{3})}}$

Cancel the common factor of $3$ .

Factor $3$ out of $18$ .

$\frac{y^{28}}{256 x^{3 (6) \frac{4}{3}}}$

Cancel the common factor.

$\frac{y^{28}}{256 x^{3 \cdot 6 \frac{4}{3}}}$

Rewrite the expression.

$\frac{y^{28}}{256 x^{6 \cdot 4}}$

Multiply $6$ by $4$ .

$\frac{y^{28}}{256 x^{24}}$

Simplify the Radical Expression (64x^18y^-21)^(-4/3)

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