Simplify ((3x^2y^-3)^3)/(54x^-8y^-10)

$\frac{{(3 x^{2} y^{- 3})}^{3}}{54 x^{- 8} y^{- 10}}$

Step 1

Simplify.

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

$\frac{{(3 x^{2} y^{- 3})}^{3} x^{8}}{54 y^{- 10}}$

Move $y^{- 10}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(3 x^{2} y^{- 3})}^{3} x^{8} y^{10}}{54}$

Step 2

Simplify the numerator.

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

$\frac{{(3 x^{2})}^{3} {(y^{- 3})}^{3} x^{8} y^{10}}{54}$

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

$\frac{3^{3} {(x^{2})}^{3} {(y^{- 3})}^{3} x^{8} y^{10}}{54}$

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

$\frac{27 {(x^{2})}^{3} {(y^{- 3})}^{3} x^{8} y^{10}}{54}$

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

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

$\frac{27 x^{2 \cdot 3} {(y^{- 3})}^{3} x^{8} y^{10}}{54}$

Multiply $2$ by $3$ .

$\frac{27 x^{6} {(y^{- 3})}^{3} x^{8} y^{10}}{54}$

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

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

$\frac{27 x^{6} y^{- 3 \cdot 3} x^{8} y^{10}}{54}$

Multiply $- 3$ by $3$ .

$\frac{27 x^{6} y^{- 9} x^{8} y^{10}}{54}$

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

$\frac{27 x^{6} \frac{1}{y^{9}} x^{8} y^{10}}{54}$

Combine exponents.

Combine $27$ and $\frac{1}{y^{9}}$ .

$\frac{x^{6} \frac{27}{y^{9}} x^{8} y^{10}}{54}$

Combine $x^{6}$ and $\frac{27}{y^{9}}$ .

$\frac{\frac{x^{6} \cdot 27}{y^{9}} x^{8} y^{10}}{54}$

Combine $\frac{x^{6} \cdot 27}{y^{9}}$ and $x^{8}$ .

$\frac{\frac{x^{6} \cdot 27 x^{8}}{y^{9}} y^{10}}{54}$

Multiply $x^{6}$ by $x^{8}$ by adding the exponents.

Move $x^{8}$ .

$\frac{\frac{x^{8} x^{6} \cdot 27}{y^{9}} y^{10}}{54}$

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

$\frac{\frac{x^{8 + 6} \cdot 27}{y^{9}} y^{10}}{54}$

Add $8$ and $6$ .

$\frac{\frac{x^{14} \cdot 27}{y^{9}} y^{10}}{54}$

Combine $\frac{x^{14} \cdot 27}{y^{9}}$ and $y^{10}$ .

$\frac{\frac{x^{14} \cdot 27 y^{10}}{y^{9}}}{54}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{x^{14} \cdot 27 y^{10}}{y^{9}}$ by cancelling the common factors.

Factor $y^{9}$ out of $x^{14} \cdot 27 y^{10}$ .

$\frac{\frac{y^{9} (x^{14} \cdot 27 y)}{y^{9}}}{54}$

Multiply by $1$ .

$\frac{\frac{y^{9} (x^{14} \cdot 27 y)}{y^{9} \cdot 1}}{54}$

Cancel the common factor.

$\frac{\frac{y^{9} (x^{14} \cdot 27 y)}{y^{9} \cdot 1}}{54}$

Rewrite the expression.

$\frac{\frac{x^{14} \cdot 27 y}{1}}{54}$

Divide $x^{14} \cdot 27 y$ by $1$ .

$\frac{x^{14} \cdot 27 y}{54}$

Step 3

Cancel the common factor of $27$ and $54$ .

Factor $27$ out of $x^{14} \cdot 27 y$ .

$\frac{27 (x^{14} y)}{54}$

Cancel the common factors.

Factor $27$ out of $54$ .

$\frac{27 (x^{14} y)}{27 \cdot 2}$

Cancel the common factor.

$\frac{27 (x^{14} y)}{27 \cdot 2}$

Rewrite the expression.

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

Simplify ((3x^2y^-3)^3)/(54x^-8y^-10)

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