Trigonometry Examples

$\frac{27 x^{17} y^{18}}{{(x^{5} y^{6})}^{\frac{1}{3}}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{27 x^{17} y^{18}}{{(x^{5})}^{\frac{1}{3}} {(y^{6})}^{\frac{1}{3}}}$

Step 1.2

Multiply the exponents in ${(x^{5})}^{\frac{1}{3}}$ .

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Step 1.2.1

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

$\frac{27 x^{17} y^{18}}{x^{5 (\frac{1}{3})} {(y^{6})}^{\frac{1}{3}}}$

Step 1.2.2

Combine $5$ and $\frac{1}{3}$ .

$\frac{27 x^{17} y^{18}}{x^{\frac{5}{3}} {(y^{6})}^{\frac{1}{3}}}$

Step 1.3

Multiply the exponents in ${(y^{6})}^{\frac{1}{3}}$ .

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Step 1.3.1

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

$\frac{27 x^{17} y^{18}}{x^{\frac{5}{3}} y^{6 (\frac{1}{3})}}$

Step 1.3.2

Cancel the common factor of $3$ .

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Step 1.3.2.1

Factor $3$ out of $6$ .

$\frac{27 x^{17} y^{18}}{x^{\frac{5}{3}} y^{3 (2) \frac{1}{3}}}$

Step 1.3.2.2

Cancel the common factor.

$\frac{27 x^{17} y^{18}}{x^{\frac{5}{3}} y^{3 \cdot 2 \frac{1}{3}}}$

Step 1.3.2.3

Rewrite the expression.

$\frac{27 x^{17} y^{18}}{x^{\frac{5}{3}} y^{2}}$

Step 2

Move $x^{\frac{5}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{27 x^{17} y^{18} x^{- \frac{5}{3}}}{y^{2}}$

Step 3

Multiply $x^{17}$ by $x^{- \frac{5}{3}}$ by adding the exponents.

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Step 3.1

Move $x^{- \frac{5}{3}}$ .

$\frac{27 (x^{- \frac{5}{3}} x^{17}) y^{18}}{y^{2}}$

Step 3.2

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

$\frac{27 x^{- \frac{5}{3} + 17} y^{18}}{y^{2}}$

Step 3.3

To write $17$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{27 x^{- \frac{5}{3} + 17 \cdot \frac{3}{3}} y^{18}}{y^{2}}$

Step 3.4

Combine $17$ and $\frac{3}{3}$ .

$\frac{27 x^{- \frac{5}{3} + \frac{17 \cdot 3}{3}} y^{18}}{y^{2}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{27 x^{\frac{- 5 + 17 \cdot 3}{3}} y^{18}}{y^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $17$ by $3$ .

$\frac{27 x^{\frac{- 5 + 51}{3}} y^{18}}{y^{2}}$

Step 3.6.2

Add $- 5$ and $51$ .

$\frac{27 x^{\frac{46}{3}} y^{18}}{y^{2}}$

Step 4

Factor $y^{2}$ out of $27 x^{\frac{46}{3}} y^{18}$ .

$\frac{y^{2} (27 x^{\frac{46}{3}} y^{16})}{y^{2}}$

Step 5

Cancel the common factors.

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Step 5.1

Multiply by $1$ .

$\frac{y^{2} (27 x^{\frac{46}{3}} y^{16})}{y^{2} \cdot 1}$

Step 5.2

Cancel the common factor.

$\frac{y^{2} (27 x^{\frac{46}{3}} y^{16})}{y^{2} \cdot 1}$

Step 5.3

Rewrite the expression.

$\frac{27 x^{\frac{46}{3}} y^{16}}{1}$

Step 5.4

Divide $27 x^{\frac{46}{3}} y^{16}$ by $1$ .

$27 x^{\frac{46}{3}} y^{16}$

Step 6

Reorder factors in $27 x^{\frac{46}{3}} y^{16}$ .

$27 y^{16} x^{\frac{46}{3}}$