Simplify ((x^2y)^(-3/2))/((x^3y^4)^(-2/3))

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

Step 1

Simplify the expression.

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

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

Move ${(x^{3} y^{4})}^{\frac{- 2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

Move the negative in front of the fraction.

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

Multiply $- - \frac{2}{3}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{2}{3}$ by $1$ .

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

Apply the product rule to $x^{3} y^{4}$ .

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the exponents in ${(y^{4})}^{\frac{2}{3}}$ .

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

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

Multiply $4 (\frac{2}{3})$ .

Combine $4$ and $\frac{2}{3}$ .

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

Multiply $4$ by $2$ .

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

Step 3

Simplify the denominator.

Move the negative in front of the fraction.

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

Multiply $- - \frac{3}{2}$ .

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} y^{\frac{8}{3}}}{{(x^{2} y)}^{1 (\frac{3}{2})}}$

Multiply $\frac{3}{2}$ by $1$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

$\frac{x^{2} y^{\frac{8}{3}} y^{- \frac{3}{2}}}{x^{3}}$

Step 5

Multiply $y^{\frac{8}{3}}$ by $y^{- \frac{3}{2}}$ by adding the exponents.

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

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

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

$\frac{x^{2} y^{- \frac{3}{2} + \frac{8}{3}}}{x^{3}}$

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

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

To write $\frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

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

Multiply $2$ by $3$ .

$\frac{x^{2} y^{- \frac{3 \cdot 3}{6} + \frac{8}{3} \cdot \frac{2}{2}}}{x^{3}}$

Multiply $\frac{8}{3}$ by $\frac{2}{2}$ .

$\frac{x^{2} y^{- \frac{3 \cdot 3}{6} + \frac{8 \cdot 2}{3 \cdot 2}}}{x^{3}}$

Multiply $3$ by $2$ .

$\frac{x^{2} y^{- \frac{3 \cdot 3}{6} + \frac{8 \cdot 2}{6}}}{x^{3}}$

Combine the numerators over the common denominator.

$\frac{x^{2} y^{\frac{- 3 \cdot 3 + 8 \cdot 2}{6}}}{x^{3}}$

Simplify the numerator.

Multiply $- 3$ by $3$ .

$\frac{x^{2} y^{\frac{- 9 + 8 \cdot 2}{6}}}{x^{3}}$

Multiply $8$ by $2$ .

$\frac{x^{2} y^{\frac{- 9 + 16}{6}}}{x^{3}}$

Add $- 9$ and $16$ .

$\frac{x^{2} y^{\frac{7}{6}}}{x^{3}}$

Step 6

Factor $x^{2}$ out of $x^{2} y^{\frac{7}{6}}$ .

$\frac{x^{2} (y^{\frac{7}{6}})}{x^{3}}$

Step 7

Cancel the common factors.

Factor $x^{2}$ out of $x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{y^{\frac{7}{6}}}{x}$

Simplify ((x^2y)^(-3/2))/((x^3y^4)^(-2/3))

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