Evaluate ((3x^(3/2)y^3)/(x^2y^(-1/2)))^-2

${(\frac{3 x^{\frac{3}{2}} y^{3}}{x^{2} y^{- \frac{1}{2}}})}^{- 2}$

Step 1

Move $x^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

${(\frac{3 y^{3}}{x^{2} y^{- \frac{1}{2}} x^{- \frac{3}{2}}})}^{- 2}$

Step 2

Multiply $x^{2}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

Move $x^{- \frac{3}{2}}$ .

${(\frac{3 y^{3}}{x^{- \frac{3}{2}} x^{2} y^{- \frac{1}{2}}})}^{- 2}$

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

${(\frac{3 y^{3}}{x^{- \frac{3}{2} + 2} y^{- \frac{1}{2}}})}^{- 2}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{3 y^{3}}{x^{- \frac{3}{2} + 2 \cdot \frac{2}{2}} y^{- \frac{1}{2}}})}^{- 2}$

Combine $2$ and $\frac{2}{2}$ .

${(\frac{3 y^{3}}{x^{- \frac{3}{2} + \frac{2 \cdot 2}{2}} y^{- \frac{1}{2}}})}^{- 2}$

Combine the numerators over the common denominator.

${(\frac{3 y^{3}}{x^{\frac{- 3 + 2 \cdot 2}{2}} y^{- \frac{1}{2}}})}^{- 2}$

Simplify the numerator.

Multiply $2$ by $2$ .

${(\frac{3 y^{3}}{x^{\frac{- 3 + 4}{2}} y^{- \frac{1}{2}}})}^{- 2}$

Add $- 3$ and $4$ .

${(\frac{3 y^{3}}{x^{\frac{1}{2}} y^{- \frac{1}{2}}})}^{- 2}$

Step 3

Move $y^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 y^{3} y^{\frac{1}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Step 4

Multiply $y^{3}$ by $y^{\frac{1}{2}}$ by adding the exponents.

Move $y^{\frac{1}{2}}$ .

${(\frac{3 (y^{\frac{1}{2}} y^{3})}{x^{\frac{1}{2}}})}^{- 2}$

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

${(\frac{3 y^{\frac{1}{2} + 3}}{x^{\frac{1}{2}}})}^{- 2}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{3 y^{\frac{1}{2} + 3 \cdot \frac{2}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Combine $3$ and $\frac{2}{2}$ .

${(\frac{3 y^{\frac{1}{2} + \frac{3 \cdot 2}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Combine the numerators over the common denominator.

${(\frac{3 y^{\frac{1 + 3 \cdot 2}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Simplify the numerator.

Multiply $3$ by $2$ .

${(\frac{3 y^{\frac{1 + 6}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Add $1$ and $6$ .

${(\frac{3 y^{\frac{7}{2}}}{x^{\frac{1}{2}}})}^{- 2}$

Step 5

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

${(\frac{x^{\frac{1}{2}}}{3 y^{\frac{7}{2}}})}^{2}$

Step 6

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

Apply the product rule to $\frac{x^{\frac{1}{2}}}{3 y^{\frac{7}{2}}}$ .

$\frac{{(x^{\frac{1}{2}})}^{2}}{{(3 y^{\frac{7}{2}})}^{2}}$

Apply the product rule to $3 y^{\frac{7}{2}}$ .

$\frac{{(x^{\frac{1}{2}})}^{2}}{3^{2} {(y^{\frac{7}{2}})}^{2}}$

Step 7

Simplify the numerator.

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$\frac{x^{\frac{1}{2} \cdot 2}}{3^{2} {(y^{\frac{7}{2}})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x^{\frac{1}{2} \cdot 2}}{3^{2} {(y^{\frac{7}{2}})}^{2}}$

Rewrite the expression.

$\frac{x^{1}}{3^{2} {(y^{\frac{7}{2}})}^{2}}$

Simplify.

$\frac{x}{3^{2} {(y^{\frac{7}{2}})}^{2}}$

Step 8

Simplify the denominator.

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

$\frac{x}{9 {(y^{\frac{7}{2}})}^{2}}$

Multiply the exponents in ${(y^{\frac{7}{2}})}^{2}$ .

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

$\frac{x}{9 y^{\frac{7}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{9 y^{\frac{7}{2} \cdot 2}}$

Rewrite the expression.

$\frac{x}{9 y^{7}}$

Evaluate ((3x^(3/2)y^3)/(x^2y^(-1/2)))^-2

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