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

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

Step 1

Simplify the expression.

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

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

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

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

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

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

Step 2

Simplify the numerator.

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

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

Apply the product rule to $x^{- 3} y^{2}$ .

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

Multiply ${(y^{2})}^{2}$ by ${(y^{2})}^{3}$ by adding the exponents.

Move ${(y^{2})}^{3}$ .

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

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

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

Add $3$ and $2$ .

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

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

$\frac{9 {(y^{2})}^{5} {(x^{- 3})}^{3}}{2^{3} x}$

Multiply the exponents in ${(y^{2})}^{5}$ .

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

$\frac{9 y^{2 \cdot 5} {(x^{- 3})}^{3}}{2^{3} x}$

Multiply $2$ by $5$ .

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

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

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

$\frac{9 y^{10} x^{- 3 \cdot 3}}{2^{3} x}$

Multiply $- 3$ by $3$ .

$\frac{9 y^{10} x^{- 9}}{2^{3} x}$

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

$\frac{9 y^{10} \frac{1}{x^{9}}}{2^{3} x}$

Combine exponents.

Combine $9$ and $\frac{1}{x^{9}}$ .

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

Combine $y^{10}$ and $\frac{9}{x^{9}}$ .

$\frac{\frac{y^{10} \cdot 9}{x^{9}}}{2^{3} x}$

Move $9$ to the left of $y^{10}$ .

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

Step 3

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

$\frac{\frac{9 y^{10}}{x^{9}}}{8 x}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 y^{10}}{x^{9}} \cdot \frac{1}{8 x}$

Step 5

Combine.

$\frac{9 y^{10} \cdot 1}{x^{9} (8 x)}$

Step 6

Multiply $x^{9}$ by $x$ by adding the exponents.

Move $x$ .

$\frac{9 y^{10} \cdot 1}{x \cdot x^{9} \cdot 8}$

Multiply $x$ by $x^{9}$ .

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

$\frac{9 y^{10} \cdot 1}{x^{1} x^{9} \cdot 8}$

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

$\frac{9 y^{10} \cdot 1}{x^{1 + 9} \cdot 8}$

Add $1$ and $9$ .

$\frac{9 y^{10} \cdot 1}{x^{10} \cdot 8}$

Step 7

Multiply $9 y^{10}$ by $1$ .

$\frac{9 y^{10}}{x^{10} \cdot 8}$

Step 8

Move $8$ to the left of $x^{10}$ .

$\frac{9 y^{10}}{8 x^{10}}$

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

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