Algebra Examples

${(\frac{x^{5} y^{- 1}}{6 y^{2}})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 1

Move $y^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{x^{5}}{6 y^{2} y})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 2

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.1

Move $y$ .

${(\frac{x^{5}}{6 (y \cdot y^{2})})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 2.2

Multiply $y$ by $y^{2}$ .

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Step 2.2.1

Raise $y$ to the power of $1$ .

${(\frac{x^{5}}{6 (y^{1} y^{2})})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 2.2.2

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

${(\frac{x^{5}}{6 y^{1 + 2}})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 2.3

Add $1$ and $2$ .

${(\frac{x^{5}}{6 y^{3}})}^{- 2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{6 y^{3}}{x^{5}})}^{2} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{6 y^{3}}{x^{5}}$ .

$\frac{{(6 y^{3})}^{2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 4.2

Apply the product rule to $6 y^{3}$ .

$\frac{6^{2} {(y^{3})}^{2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $6$ to the power of $2$ .

$\frac{36 {(y^{3})}^{2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 5.2

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

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Step 5.2.1

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

$\frac{36 y^{3 \cdot 2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 5.2.2

Multiply $3$ by $2$ .

$\frac{36 y^{6}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 6

Simplify the expression.

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Step 6.1

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

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Step 6.1.1

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

$\frac{36 y^{6}}{x^{5 \cdot 2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 6.1.2

Multiply $5$ by $2$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 6.2

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y \cdot y^{2}})}^{2}$

Step 7

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 7.1

Move $y^{2}$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 (y^{2} y)})}^{2}$

Step 7.2

Multiply $y^{2}$ by $y$ .

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Step 7.2.1

Raise $y$ to the power of $1$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 (y^{2} y^{1})})}^{2}$

Step 7.2.2

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

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y^{2 + 1}})}^{2}$

Step 7.3

Add $2$ and $1$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y^{3}})}^{2}$

Step 8

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

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Step 8.1

Apply the product rule to $\frac{x^{2}}{9 y^{3}}$ .

$\frac{36 y^{6}}{x^{10}} \cdot \frac{{(x^{2})}^{2}}{{(9 y^{3})}^{2}}$

Step 8.2

Apply the product rule to $9 y^{3}$ .

$\frac{36 y^{6}}{x^{10}} \cdot \frac{{(x^{2})}^{2}}{9^{2} {(y^{3})}^{2}}$

Step 9

Combine fractions.

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Step 9.1

Combine.

$\frac{36 y^{6} {(x^{2})}^{2}}{x^{10} (9^{2} {(y^{3})}^{2})}$

Step 9.2

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

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Step 9.2.1

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

$\frac{36 y^{6} x^{2 \cdot 2}}{x^{10} \cdot 9^{2} {(y^{3})}^{2}}$

Step 9.2.2

Multiply $2$ by $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 9^{2} {(y^{3})}^{2}}$

Step 10

Simplify the denominator.

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Step 10.1

Raise $9$ to the power of $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 {(y^{3})}^{2}}$

Step 10.2

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

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Step 10.2.1

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

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 y^{3 \cdot 2}}$

Step 10.2.2

Multiply $3$ by $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 y^{6}}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Cancel the common factor of $36$ and $81$ .

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Step 11.1.1

Factor $9$ out of $36 y^{6} x^{4}$ .

$\frac{9 (4 y^{6} x^{4})}{x^{10} \cdot 81 y^{6}}$

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

Factor $9$ out of $x^{10} \cdot 81 y^{6}$ .

$\frac{9 (4 y^{6} x^{4})}{9 (x^{10} \cdot 9 y^{6})}$

Step 11.1.2.2

Cancel the common factor.

$\frac{9 (4 y^{6} x^{4})}{9 (x^{10} \cdot 9 y^{6})}$

Step 11.1.2.3

Rewrite the expression.

$\frac{4 y^{6} x^{4}}{x^{10} \cdot 9 y^{6}}$

Step 11.2

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

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Step 11.2.1

Cancel the common factor.

$\frac{4 y^{6} x^{4}}{x^{10} \cdot 9 y^{6}}$

Step 11.2.2

Rewrite the expression.

$\frac{4 x^{4}}{x^{10} \cdot 9}$

Step 11.3

Cancel the common factor of $x^{4}$ and $x^{10}$ .

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Step 11.3.1

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

$\frac{x^{4} \cdot 4}{x^{10} \cdot 9}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $x^{4}$ out of $x^{10} \cdot 9$ .

$\frac{x^{4} \cdot 4}{x^{4} (x^{6} \cdot 9)}$

Step 11.3.2.2

Cancel the common factor.

$\frac{x^{4} \cdot 4}{x^{4} (x^{6} \cdot 9)}$

Step 11.3.2.3

Rewrite the expression.

$\frac{4}{x^{6} \cdot 9}$

Step 11.4

Move $9$ to the left of $x^{6}$ .

$\frac{4}{9 x^{6}}$