Precalculus Examples

$(3 x^{3} y^{4}) {(2 x^{- 5} y^{- 2})}^{2}$

Step 1

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

$3 x^{3} y^{4} {(2 \frac{1}{x^{5}} y^{- 2})}^{2}$

Step 2

Combine $2$ and $\frac{1}{x^{5}}$ .

$3 x^{3} y^{4} {(\frac{2}{x^{5}} y^{- 2})}^{2}$

Step 3

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

$3 x^{3} y^{4} {(\frac{2}{x^{5}} \cdot \frac{1}{y^{2}})}^{2}$

Step 4

Combine.

$3 x^{3} y^{4} {(\frac{2 \cdot 1}{x^{5} y^{2}})}^{2}$

Step 5

Multiply $2$ by $1$ .

$3 x^{3} y^{4} {(\frac{2}{x^{5} y^{2}})}^{2}$

Step 6

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

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Step 6.1

Apply the product rule to $\frac{2}{x^{5} y^{2}}$ .

$3 x^{3} y^{4} \frac{2^{2}}{{(x^{5} y^{2})}^{2}}$

Step 6.2

Apply the product rule to $x^{5} y^{2}$ .

$3 x^{3} y^{4} \frac{2^{2}}{{(x^{5})}^{2} {(y^{2})}^{2}}$

Step 7

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

$3 x^{3} y^{4} \frac{4}{{(x^{5})}^{2} {(y^{2})}^{2}}$

Step 8

Simplify the denominator.

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Step 8.1

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

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Step 8.1.1

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

$3 x^{3} y^{4} \frac{4}{x^{5 \cdot 2} {(y^{2})}^{2}}$

Step 8.1.2

Multiply $5$ by $2$ .

$3 x^{3} y^{4} \frac{4}{x^{10} {(y^{2})}^{2}}$

Step 8.2

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

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Step 8.2.1

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

$3 x^{3} y^{4} \frac{4}{x^{10} y^{2 \cdot 2}}$

Step 8.2.2

Multiply $2$ by $2$ .

$3 x^{3} y^{4} \frac{4}{x^{10} y^{4}}$

Step 9

Simplify terms.

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Step 9.1

Cancel the common factor of $x^{3} y^{4}$ .

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Step 9.1.1

Factor $x^{3} y^{4}$ out of $3 x^{3} y^{4}$ .

$x^{3} y^{4} (3) \frac{4}{x^{10} y^{4}}$

Step 9.1.2

Factor $x^{3} y^{4}$ out of $x^{10} y^{4}$ .

$x^{3} y^{4} (3) \frac{4}{x^{3} y^{4} (x^{7})}$

Step 9.1.3

Cancel the common factor.

$x^{3} y^{4} \cdot 3 \frac{4}{x^{3} y^{4} x^{7}}$

Step 9.1.4

Rewrite the expression.

$3 \frac{4}{x^{7}}$

Step 9.2

Combine $3$ and $\frac{4}{x^{7}}$ .

$\frac{3 \cdot 4}{x^{7}}$

Step 9.3

Multiply $3$ by $4$ .

$\frac{12}{x^{7}}$