Algebra Examples

$\frac{{(3 x^{2} y^{- 3})}^{3}}{27 {(x y)}^{- 9}}$

Step 1

Move ${(x y)}^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(3 x^{2} y^{- 3})}^{3} {(x y)}^{9}}{27}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(3 x^{2})}^{3} {(y^{- 3})}^{3} {(x y)}^{9}}{27}$

Step 2.2

Apply the product rule to $x y$ .

$\frac{{(3 x^{2})}^{3} {(y^{- 3})}^{3} x^{9} y^{9}}{27}$

Step 2.3

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

$\frac{3^{3} {(x^{2})}^{3} {(y^{- 3})}^{3} x^{9} y^{9}}{27}$

Step 2.4

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

$\frac{27 {(x^{2})}^{3} {(y^{- 3})}^{3} x^{9} y^{9}}{27}$

Step 2.5

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

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Step 2.5.1

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

$\frac{27 x^{2 \cdot 3} {(y^{- 3})}^{3} x^{9} y^{9}}{27}$

Step 2.5.2

Multiply $2$ by $3$ .

$\frac{27 x^{6} {(y^{- 3})}^{3} x^{9} y^{9}}{27}$

Step 2.6

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

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Step 2.6.1

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

$\frac{27 x^{6} y^{- 3 \cdot 3} x^{9} y^{9}}{27}$

Step 2.6.2

Multiply $- 3$ by $3$ .

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

Step 2.7

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

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

Step 2.8

Combine exponents.

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Step 2.8.1

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

$\frac{x^{6} \frac{27}{y^{9}} x^{9} y^{9}}{27}$

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.8.4

Multiply $x^{6}$ by $x^{9}$ by adding the exponents.

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Step 2.8.4.1

Move $x^{9}$ .

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

Step 2.8.4.2

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

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

Step 2.8.4.3

Add $9$ and $6$ .

$\frac{\frac{x^{15} \cdot 27}{y^{9}} y^{9}}{27}$

Step 2.8.5

Combine $\frac{x^{15} \cdot 27}{y^{9}}$ and $y^{9}$ .

$\frac{\frac{x^{15} \cdot 27 y^{9}}{y^{9}}}{27}$

Step 2.9

Reduce the expression by cancelling the common factors.

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Step 2.9.1

Reduce the expression $\frac{x^{15} \cdot 27 y^{9}}{y^{9}}$ by cancelling the common factors.

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Step 2.9.1.1

Cancel the common factor.

$\frac{\frac{x^{15} \cdot 27 y^{9}}{y^{9}}}{27}$

Step 2.9.1.2

Rewrite the expression.

$\frac{\frac{x^{15} \cdot 27}{1}}{27}$

Step 2.9.2

Divide $x^{15} \cdot 27$ by $1$ .

$\frac{x^{15} \cdot 27}{27}$

Step 3

Cancel the common factor of $27$ .

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Step 3.1

Cancel the common factor.

$\frac{x^{15} \cdot 27}{27}$

Step 3.2

Divide $x^{15}$ by $1$ .

$x^{15}$