Simplify ((3x^2y^5)/(2x^-1y^-2))^2

${(\frac{3 x^{2} y^{5}}{2 x^{- 1} y^{- 2}})}^{2}$

Step 1

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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

Move $x$ .

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

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

${(\frac{3 (x^{1} x^{2}) y^{5} y^{2}}{2})}^{2}$

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

${(\frac{3 x^{1 + 2} y^{5} y^{2}}{2})}^{2}$

Add $1$ and $2$ .

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

Multiply $y^{5}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

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

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

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

Add $2$ and $5$ .

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

Step 4

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

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

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

Apply the product rule to $3 x^{3} y^{7}$ .

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

Apply the product rule to $3 x^{3}$ .

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

Step 5

Simplify the numerator.

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

$\frac{9 {(x^{3})}^{2} {(y^{7})}^{2}}{2^{2}}$

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

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

$\frac{9 x^{3 \cdot 2} {(y^{7})}^{2}}{2^{2}}$

Multiply $3$ by $2$ .

$\frac{9 x^{6} {(y^{7})}^{2}}{2^{2}}$

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

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

$\frac{9 x^{6} y^{7 \cdot 2}}{2^{2}}$

Multiply $7$ by $2$ .

$\frac{9 x^{6} y^{14}}{2^{2}}$

Step 6

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

$\frac{9 x^{6} y^{14}}{4}$

Simplify ((3x^2y^5)/(2x^-1y^-2))^2

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