Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $3$ .

$8 x^{6} {(y^{3})}^{3} {(3 x^{3} y)}^{- 2}$

Step 4

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

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Step 4.1

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

$8 x^{6} y^{3 \cdot 3} {(3 x^{3} y)}^{- 2}$

Step 4.2

Multiply $3$ by $3$ .

$8 x^{6} y^{9} {(3 x^{3} y)}^{- 2}$

Step 5

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

$8 x^{6} y^{9} \frac{1}{{(3 x^{3} y)}^{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

$8 x^{6} y^{9} \frac{1}{{(3 x^{3})}^{2} y^{2}}$

Step 6.2

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

$8 x^{6} y^{9} \frac{1}{3^{2} {(x^{3})}^{2} y^{2}}$

Step 6.3

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

$8 x^{6} y^{9} \frac{1}{9 {(x^{3})}^{2} y^{2}}$

Step 6.4

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

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Step 6.4.1

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

$8 x^{6} y^{9} \frac{1}{9 x^{3 \cdot 2} y^{2}}$

Step 6.4.2

Multiply $3$ by $2$ .

$8 x^{6} y^{9} \frac{1}{9 x^{6} y^{2}}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x^{6} y^{2}$ .

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Step 7.1.1

Factor $x^{6} y^{2}$ out of $8 x^{6} y^{9}$ .

$x^{6} y^{2} (8 y^{7}) \frac{1}{9 x^{6} y^{2}}$

Step 7.1.2

Factor $x^{6} y^{2}$ out of $9 x^{6} y^{2}$ .

$x^{6} y^{2} (8 y^{7}) \frac{1}{x^{6} y^{2} (9)}$

Step 7.1.3

Cancel the common factor.

$x^{6} y^{2} (8 y^{7}) \frac{1}{x^{6} y^{2} \cdot 9}$

Step 7.1.4

Rewrite the expression.

$8 y^{7} \frac{1}{9}$

Step 7.2

Combine $8$ and $\frac{1}{9}$ .

$y^{7} \frac{8}{9}$

Step 7.3

Combine $y^{7}$ and $\frac{8}{9}$ .

$\frac{y^{7} \cdot 8}{9}$

Step 7.4

Move $8$ to the left of $y^{7}$ .

$\frac{8 y^{7}}{9}$