Calculus Examples

$\frac{y^{- \frac{1}{3}}}{y^{\frac{1}{2}} y^{\frac{4}{3}}}$

Step 1

Multiply $y^{\frac{1}{2}}$ by $y^{\frac{4}{3}}$ by adding the exponents.

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Step 1.1

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

$\frac{y^{- \frac{1}{3}}}{y^{\frac{1}{2} + \frac{4}{3}}}$

Step 1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3}}}$

Step 1.3

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3} \cdot \frac{2}{2}}}$

Step 1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3}{2 \cdot 3} + \frac{4}{3} \cdot \frac{2}{2}}}$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3}{6} + \frac{4}{3} \cdot \frac{2}{2}}}$

Step 1.4.3

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3}{6} + \frac{4 \cdot 2}{3 \cdot 2}}}$

Step 1.4.4

Multiply $3$ by $2$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3}{6} + \frac{4 \cdot 2}{6}}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3 + 4 \cdot 2}{6}}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $4$ by $2$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{3 + 8}{6}}}$

Step 1.6.2

Add $3$ and $8$ .

$\frac{y^{- \frac{1}{3}}}{y^{\frac{11}{6}}}$

Step 2

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

$\frac{1}{y^{\frac{11}{6}} y^{\frac{1}{3}}}$

Step 3

Multiply $y^{\frac{11}{6}}$ by $y^{\frac{1}{3}}$ by adding the exponents.

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Step 3.1

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

$\frac{1}{y^{\frac{11}{6} + \frac{1}{3}}}$

Step 3.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{y^{\frac{11}{6} + \frac{1}{3} \cdot \frac{2}{2}}}$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{1}{y^{\frac{11}{6} + \frac{2}{3 \cdot 2}}}$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{1}{y^{\frac{11}{6} + \frac{2}{6}}}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{y^{\frac{11 + 2}{6}}}$

Step 3.5

Add $11$ and $2$ .

$\frac{1}{y^{\frac{13}{6}}}$