Trigonometry Examples

$27 x^{\frac{3}{4}} {(16 x y^{- 2})}^{- \frac{1}{4}}$

Step 1

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

$27 x^{\frac{3}{4}} {(16 x \frac{1}{y^{2}})}^{- \frac{1}{4}}$

Step 2

Multiply $16 x \frac{1}{y^{2}}$ .

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Step 2.1

Combine $16$ and $\frac{1}{y^{2}}$ .

$27 x^{\frac{3}{4}} {(x \frac{16}{y^{2}})}^{- \frac{1}{4}}$

Step 2.2

Combine $x$ and $\frac{16}{y^{2}}$ .

$27 x^{\frac{3}{4}} {(\frac{x \cdot 16}{y^{2}})}^{- \frac{1}{4}}$

Step 3

Move $16$ to the left of $x$ .

$27 x^{\frac{3}{4}} {(\frac{16 \cdot x}{y^{2}})}^{- \frac{1}{4}}$

Step 4

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

$27 x^{\frac{3}{4}} {(\frac{y^{2}}{16 x})}^{\frac{1}{4}}$

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 $\frac{y^{2}}{16 x}$ .

$27 x^{\frac{3}{4}} \frac{{(y^{2})}^{\frac{1}{4}}}{{(16 x)}^{\frac{1}{4}}}$

Step 5.2

Apply the product rule to $16 x$ .

$27 x^{\frac{3}{4}} \frac{{(y^{2})}^{\frac{1}{4}}}{16^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 6

Multiply the exponents in ${(y^{2})}^{\frac{1}{4}}$ .

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Step 6.1

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

$27 x^{\frac{3}{4}} \frac{y^{2 (\frac{1}{4})}}{16^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

$27 x^{\frac{3}{4}} \frac{y^{2 \frac{1}{2 (2)}}}{16^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 6.2.2

Cancel the common factor.

$27 x^{\frac{3}{4}} \frac{y^{2 \frac{1}{2 \cdot 2}}}{16^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 6.2.3

Rewrite the expression.

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{16^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $16$ as $2^{4}$ .

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{{(2^{4})}^{\frac{1}{4}} x^{\frac{1}{4}}}$

Step 7.2

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

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{2^{4 (\frac{1}{4})} x^{\frac{1}{4}}}$

Step 7.3

Cancel the common factor of $4$ .

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Step 7.3.1

Cancel the common factor.

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{2^{4 (\frac{1}{4})} x^{\frac{1}{4}}}$

Step 7.3.2

Rewrite the expression.

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{2^{1} x^{\frac{1}{4}}}$

Step 7.4

Evaluate the exponent.

$27 x^{\frac{3}{4}} \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{4}}}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $x^{\frac{1}{4}}$ .

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Step 8.1.1

Factor $x^{\frac{1}{4}}$ out of $27 x^{\frac{3}{4}}$ .

$x^{\frac{1}{4}} (27 x^{\frac{2}{4}}) \frac{y^{\frac{1}{2}}}{2 x^{\frac{1}{4}}}$

Step 8.1.2

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

$x^{\frac{1}{4}} (27 x^{\frac{2}{4}}) \frac{y^{\frac{1}{2}}}{x^{\frac{1}{4}} \cdot 2}$

Step 8.1.3

Cancel the common factor.

$x^{\frac{1}{4}} (27 x^{\frac{2}{4}}) \frac{y^{\frac{1}{2}}}{x^{\frac{1}{4}} \cdot 2}$

Step 8.1.4

Rewrite the expression.

$27 x^{\frac{2}{4}} \frac{y^{\frac{1}{2}}}{2}$

Step 8.2

Combine $27$ and $\frac{y^{\frac{1}{2}}}{2}$ .

$x^{\frac{2}{4}} \frac{27 y^{\frac{1}{2}}}{2}$

Step 8.3

Combine $x^{\frac{2}{4}}$ and $\frac{27 y^{\frac{1}{2}}}{2}$ .

$\frac{x^{\frac{2}{4}} (27 y^{\frac{1}{2}})}{2}$

Step 8.4

Cancel the common factor of $2$ and $4$ .

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Step 8.4.1

Factor $2$ out of $2$ .

$\frac{x^{\frac{2 (1)}{4}} (27 y^{\frac{1}{2}})}{2}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $2$ out of $4$ .

$\frac{x^{\frac{2 \cdot 1}{2 \cdot 2}} (27 y^{\frac{1}{2}})}{2}$

Step 8.4.2.2

Cancel the common factor.

$\frac{x^{\frac{2 \cdot 1}{2 \cdot 2}} (27 y^{\frac{1}{2}})}{2}$

Step 8.4.2.3

Rewrite the expression.

$\frac{x^{\frac{1}{2}} (27 y^{\frac{1}{2}})}{2}$

Step 8.5

Move $27$ to the left of $x^{\frac{1}{2}}$ .

$\frac{27 x^{\frac{1}{2}} y^{\frac{1}{2}}}{2}$