Algebra Examples

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

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 $- x y$ .

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

Step 1.2

Apply the product rule to $- x$ .

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

Step 2

Raise $- 1$ to the power of $4$ .

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

Step 3

Multiply $x^{4}$ by $1$ .

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

Step 4

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

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

Step 5

Combine $- 3$ and $\frac{1}{x^{2}}$ .

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

Step 6

Move the negative in front of the fraction.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the expression.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Raise $- 1$ to the power of $3$ .

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

Step 8.3

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

$- x^{4} y^{4} \frac{27}{{(x^{2})}^{3}}$

Step 8.4

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

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Step 8.4.1

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

$- x^{4} y^{4} \frac{27}{x^{2 \cdot 3}}$

Step 8.4.2

Multiply $2$ by $3$ .

$- x^{4} y^{4} \frac{27}{x^{6}}$

Step 9

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

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Step 9.1

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

$x^{4} (- y^{4}) \frac{27}{x^{6}}$

Step 9.2

Factor $x^{4}$ out of $x^{6}$ .

$x^{4} (- y^{4}) \frac{27}{x^{4} x^{2}}$

Step 9.3

Cancel the common factor.

$x^{4} (- y^{4}) \frac{27}{x^{4} x^{2}}$

Step 9.4

Rewrite the expression.

$- y^{4} \frac{27}{x^{2}}$

Step 10

Combine $\frac{27}{x^{2}}$ and $y^{4}$ .

$- \frac{27 y^{4}}{x^{2}}$