Simplify ((36x^-8y^3)/(9x^2y))^(3/2)

${(\frac{36 x^{- 8} y^{3}}{9 x^{2} y})}^{\frac{3}{2}}$

Step 1

Move $x^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{36 y^{3}}{9 x^{2} y x^{8}})}^{\frac{3}{2}}$

Step 2

Multiply $x^{2}$ by $x^{8}$ by adding the exponents.

Move $x^{8}$ .

${(\frac{36 y^{3}}{9 (x^{8} x^{2}) y})}^{\frac{3}{2}}$

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

${(\frac{36 y^{3}}{9 x^{8 + 2} y})}^{\frac{3}{2}}$

Add $8$ and $2$ .

${(\frac{36 y^{3}}{9 x^{10} y})}^{\frac{3}{2}}$

Step 3

Cancel the common factor of $36$ and $9$ .

Factor $9$ out of $36 y^{3}$ .

${(\frac{9 (4 y^{3})}{9 x^{10} y})}^{\frac{3}{2}}$

Cancel the common factors.

Factor $9$ out of $9 x^{10} y$ .

${(\frac{9 (4 y^{3})}{9 (x^{10} y)})}^{\frac{3}{2}}$

Cancel the common factor.

${(\frac{9 (4 y^{3})}{9 (x^{10} y)})}^{\frac{3}{2}}$

Rewrite the expression.

${(\frac{4 y^{3}}{x^{10} y})}^{\frac{3}{2}}$

Step 4

Cancel the common factor of $y^{3}$ and $y$ .

Factor $y$ out of $4 y^{3}$ .

${(\frac{y (4 y^{2})}{x^{10} y})}^{\frac{3}{2}}$

Cancel the common factors.

Factor $y$ out of $x^{10} y$ .

${(\frac{y (4 y^{2})}{y x^{10}})}^{\frac{3}{2}}$

Cancel the common factor.

${(\frac{y (4 y^{2})}{y x^{10}})}^{\frac{3}{2}}$

Rewrite the expression.

${(\frac{4 y^{2}}{x^{10}})}^{\frac{3}{2}}$

Step 5

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

Apply the product rule to $\frac{4 y^{2}}{x^{10}}$ .

$\frac{{(4 y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Apply the product rule to $4 y^{2}$ .

$\frac{4^{\frac{3}{2}} {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Step 6

Simplify the numerator.

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

$\frac{{(2^{2})}^{\frac{3}{2}} {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

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

$\frac{2^{2 (\frac{3}{2})} {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2^{2 (\frac{3}{2})} {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{2^{3} {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Raise $2$ to the power of $3$ .

$\frac{8 {(y^{2})}^{\frac{3}{2}}}{{(x^{10})}^{\frac{3}{2}}}$

Multiply the exponents in ${(y^{2})}^{\frac{3}{2}}$ .

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

$\frac{8 y^{2 (\frac{3}{2})}}{{(x^{10})}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 y^{2 (\frac{3}{2})}}{{(x^{10})}^{\frac{3}{2}}}$

Rewrite the expression.

$\frac{8 y^{3}}{{(x^{10})}^{\frac{3}{2}}}$

Step 7

Multiply the exponents in ${(x^{10})}^{\frac{3}{2}}$ .

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

$\frac{8 y^{3}}{x^{10 (\frac{3}{2})}}$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{8 y^{3}}{x^{2 (5) \frac{3}{2}}}$

Cancel the common factor.

$\frac{8 y^{3}}{x^{2 \cdot 5 \frac{3}{2}}}$

Rewrite the expression.

$\frac{8 y^{3}}{x^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{8 y^{3}}{x^{15}}$

Simplify ((36x^-8y^3)/(9x^2y))^(3/2)

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