Trigonometry Examples

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $\frac{x^{2}}{3 y^{3}}$ .

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

Step 6.2

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $2$ .

$2 \frac{x^{4}}{3^{2} {(y^{3})}^{2}} x^{- 6} y^{2}$

Step 8

Simplify the denominator.

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Step 8.1

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

$2 \frac{x^{4}}{9 {(y^{3})}^{2}} x^{- 6} y^{2}$

Step 8.2

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

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Step 8.2.1

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

$2 \frac{x^{4}}{9 y^{3 \cdot 2}} x^{- 6} y^{2}$

Step 8.2.2

Multiply $3$ by $2$ .

$2 \frac{x^{4}}{9 y^{6}} x^{- 6} y^{2}$

Step 9

Combine $2$ and $\frac{x^{4}}{9 y^{6}}$ .

$\frac{2 x^{4}}{9 y^{6}} x^{- 6} y^{2}$

Step 10

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

$\frac{2 x^{4}}{9 y^{6}} \cdot \frac{1}{x^{6}} y^{2}$

Step 11

Combine.

$\frac{2 x^{4} \cdot 1}{9 y^{6} x^{6}} y^{2}$

Step 12

Cancel the common factor of $y^{2}$ .

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Step 12.1

Factor $y^{2}$ out of $9 y^{6} x^{6}$ .

$\frac{2 x^{4} \cdot 1}{y^{2} (9 y^{4} x^{6})} y^{2}$

Step 12.2

Cancel the common factor.

$\frac{2 x^{4} \cdot 1}{y^{2} (9 y^{4} x^{6})} y^{2}$

Step 12.3

Rewrite the expression.

$\frac{2 x^{4} \cdot 1}{9 y^{4} x^{6}}$

Step 13

Cancel the common factor of $x^{4}$ and $x^{6}$ .

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Step 13.1

Factor $x^{4}$ out of $2 x^{4} \cdot 1$ .

$\frac{x^{4} (2 \cdot 1)}{9 y^{4} x^{6}}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

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

$\frac{x^{4} (2 \cdot 1)}{x^{4} (9 y^{4} x^{2})}$

Step 13.2.2

Cancel the common factor.

$\frac{x^{4} (2 \cdot 1)}{x^{4} (9 y^{4} x^{2})}$

Step 13.2.3

Rewrite the expression.

$\frac{2 \cdot 1}{9 y^{4} x^{2}}$

Step 14

Multiply $2$ by $1$ .

$\frac{2}{9 y^{4} x^{2}}$