Algebra Examples

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

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 $- 2 x^{5} y$ .

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

Step 1.2

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

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

Step 2

Raise $- 2$ to the power of $4$ .

$16 {(x^{5})}^{4} y^{4} \cdot {(- 3 x^{- 4} y^{2})}^{2}$

Step 3

Multiply the exponents in ${(x^{5})}^{4}$ .

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Step 3.1

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

$16 x^{5 \cdot 4} y^{4} \cdot {(- 3 x^{- 4} y^{2})}^{2}$

Step 3.2

Multiply $5$ by $4$ .

$16 x^{20} y^{4} \cdot {(- 3 x^{- 4} y^{2})}^{2}$

Step 4

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

$16 x^{20} y^{4} \cdot {(- 3 \frac{1}{x^{4}} y^{2})}^{2}$

Step 5

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$16 x^{20} y^{4} \cdot {(\frac{- 3}{x^{4}} y^{2})}^{2}$

Step 6

Move the negative in front of the fraction.

$16 x^{20} y^{4} \cdot {(- \frac{3}{x^{4}} y^{2})}^{2}$

Step 7

Combine $y^{2}$ and $\frac{3}{x^{4}}$ .

$16 x^{20} y^{4} \cdot {(- \frac{y^{2} \cdot 3}{x^{4}})}^{2}$

Step 8

Move $3$ to the left of $y^{2}$ .

$16 x^{20} y^{4} \cdot {(- \frac{3 \cdot y^{2}}{x^{4}})}^{2}$

Step 9

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

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Step 9.1

Apply the product rule to $- \frac{3 y^{2}}{x^{4}}$ .

$16 x^{20} y^{4} \cdot ({(- 1)}^{2} {(\frac{3 y^{2}}{x^{4}})}^{2})$

Step 9.2

Apply the product rule to $\frac{3 y^{2}}{x^{4}}$ .

$16 x^{20} y^{4} \cdot ({(- 1)}^{2} \frac{{(3 y^{2})}^{2}}{{(x^{4})}^{2}})$

Step 9.3

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

$16 x^{20} y^{4} \cdot ({(- 1)}^{2} \frac{3^{2} {(y^{2})}^{2}}{{(x^{4})}^{2}})$

Step 10

Simplify the expression.

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Step 10.1

Raise $- 1$ to the power of $2$ .

$16 x^{20} y^{4} \cdot (1 \frac{3^{2} {(y^{2})}^{2}}{{(x^{4})}^{2}})$

Step 10.2

Multiply $\frac{3^{2} {(y^{2})}^{2}}{{(x^{4})}^{2}}$ by $1$ .

$16 x^{20} y^{4} \cdot \frac{3^{2} {(y^{2})}^{2}}{{(x^{4})}^{2}}$

Step 11

Simplify the numerator.

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Step 11.1

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

$16 x^{20} y^{4} \cdot \frac{9 {(y^{2})}^{2}}{{(x^{4})}^{2}}$

Step 11.2

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

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Step 11.2.1

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

$16 x^{20} y^{4} \cdot \frac{9 y^{2 \cdot 2}}{{(x^{4})}^{2}}$

Step 11.2.2

Multiply $2$ by $2$ .

$16 x^{20} y^{4} \cdot \frac{9 y^{4}}{{(x^{4})}^{2}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 12.1.1

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

$16 x^{20} y^{4} \cdot \frac{9 y^{4}}{x^{4 \cdot 2}}$

Step 12.1.2

Multiply $4$ by $2$ .

$16 x^{20} y^{4} \cdot \frac{9 y^{4}}{x^{8}}$

Step 12.2

Cancel the common factor of $x^{8}$ .

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Step 12.2.1

Factor $x^{8}$ out of $16 x^{20} y^{4}$ .

$x^{8} (16 x^{12} y^{4}) \cdot \frac{9 y^{4}}{x^{8}}$

Step 12.2.2

Cancel the common factor.

$x^{8} (16 x^{12} y^{4}) \cdot \frac{9 y^{4}}{x^{8}}$

Step 12.2.3

Rewrite the expression.

$16 x^{12} y^{4} \cdot (9 y^{4})$

Step 12.3

Multiply $9$ by $16$ .

$144 x^{12} y^{4} \cdot y^{4}$

Step 13

Multiply $y^{4}$ by $y^{4}$ by adding the exponents.

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Step 13.1

Move $y^{4}$ .

$144 x^{12} (y^{4} y^{4})$

Step 13.2

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

$144 x^{12} y^{4 + 4}$

Step 13.3

Add $4$ and $4$ .

$144 x^{12} y^{8}$