Algebra Examples

${(3 x^{- 6} y^{2})}^{3} \cdot 2 x^{10} y^{- 7}$

Step 1

Rewrite using the commutative property of multiplication.

$2 {(3 x^{- 6} y^{2})}^{3} x^{10} y^{- 7}$

Step 2

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

$2 {(3 \frac{1}{x^{6}} y^{2})}^{3} x^{10} y^{- 7}$

Step 3

Combine $3$ and $\frac{1}{x^{6}}$ .

$2 {(\frac{3}{x^{6}} y^{2})}^{3} x^{10} y^{- 7}$

Step 4

Combine $\frac{3}{x^{6}}$ and $y^{2}$ .

$2 {(\frac{3 y^{2}}{x^{6}})}^{3} x^{10} y^{- 7}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{3 y^{2}}{x^{6}}$ .

$2 \frac{{(3 y^{2})}^{3}}{{(x^{6})}^{3}} x^{10} y^{- 7}$

Step 5.2

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

$2 \frac{3^{3} {(y^{2})}^{3}}{{(x^{6})}^{3}} x^{10} y^{- 7}$

Step 6

Simplify the numerator.

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Step 6.1

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

$2 \frac{27 {(y^{2})}^{3}}{{(x^{6})}^{3}} x^{10} y^{- 7}$

Step 6.2

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

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Step 6.2.1

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

$2 \frac{27 y^{2 \cdot 3}}{{(x^{6})}^{3}} x^{10} y^{- 7}$

Step 6.2.2

Multiply $2$ by $3$ .

$2 \frac{27 y^{6}}{{(x^{6})}^{3}} x^{10} y^{- 7}$

Step 7

Multiply the exponents in ${(x^{6})}^{3}$ .

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Step 7.1

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

$2 \frac{27 y^{6}}{x^{6 \cdot 3}} x^{10} y^{- 7}$

Step 7.2

Multiply $6$ by $3$ .

$2 \frac{27 y^{6}}{x^{18}} x^{10} y^{- 7}$

Step 8

Multiply $2 \frac{27 y^{6}}{x^{18}}$ .

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Step 8.1

Combine $2$ and $\frac{27 y^{6}}{x^{18}}$ .

$\frac{2 (27 y^{6})}{x^{18}} x^{10} y^{- 7}$

Step 8.2

Multiply $27$ by $2$ .

$\frac{54 y^{6}}{x^{18}} x^{10} y^{- 7}$

Step 9

Cancel the common factor of $x^{10}$ .

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Step 9.1

Factor $x^{10}$ out of $x^{18}$ .

$\frac{54 y^{6}}{x^{10} x^{8}} x^{10} y^{- 7}$

Step 9.2

Cancel the common factor.

$\frac{54 y^{6}}{x^{10} x^{8}} x^{10} y^{- 7}$

Step 9.3

Rewrite the expression.

$\frac{54 y^{6}}{x^{8}} y^{- 7}$

Step 10

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

$\frac{54 y^{6}}{x^{8}} \cdot \frac{1}{y^{7}}$

Step 11

Combine.

$\frac{54 y^{6} \cdot 1}{x^{8} y^{7}}$

Step 12

Cancel the common factor of $y^{6}$ and $y^{7}$ .

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Step 12.1

Factor $y^{6}$ out of $54 y^{6} \cdot 1$ .

$\frac{y^{6} (54 \cdot 1)}{x^{8} y^{7}}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $y^{6}$ out of $x^{8} y^{7}$ .

$\frac{y^{6} (54 \cdot 1)}{y^{6} (x^{8} y)}$

Step 12.2.2

Cancel the common factor.

$\frac{y^{6} (54 \cdot 1)}{y^{6} (x^{8} y)}$

Step 12.2.3

Rewrite the expression.

$\frac{54 \cdot 1}{x^{8} y}$

Step 13

Multiply $54$ by $1$ .

$\frac{54}{x^{8} y}$