Algebra Examples

$\frac{2 x^{\frac{1}{2}} y^{\frac{1}{3}}}{2 x^{\frac{4}{3}} y^{- \frac{7}{4}}}$

Step 1

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{\frac{4}{3}} y^{- \frac{7}{4}} x^{- \frac{1}{2}}}$

Step 2

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

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Step 2.1

Move $x^{- \frac{1}{2}}$ .

$\frac{2 y^{\frac{1}{3}}}{2 (x^{- \frac{1}{2}} x^{\frac{4}{3}}) y^{- \frac{7}{4}}}$

Step 2.2

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{1}{2} + \frac{4}{3}} y^{- \frac{7}{4}}}$

Step 2.3

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3}} y^{- \frac{7}{4}}}$

Step 2.4

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3} \cdot \frac{2}{2}} y^{- \frac{7}{4}}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{3}{2 \cdot 3} + \frac{4}{3} \cdot \frac{2}{2}} y^{- \frac{7}{4}}}$

Step 2.5.2

Multiply $2$ by $3$ .

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{3}{6} + \frac{4}{3} \cdot \frac{2}{2}} y^{- \frac{7}{4}}}$

Step 2.5.3

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

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{3}{6} + \frac{4 \cdot 2}{3 \cdot 2}} y^{- \frac{7}{4}}}$

Step 2.5.4

Multiply $3$ by $2$ .

$\frac{2 y^{\frac{1}{3}}}{2 x^{- \frac{3}{6} + \frac{4 \cdot 2}{6}} y^{- \frac{7}{4}}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{2 y^{\frac{1}{3}}}{2 x^{\frac{- 3 + 4 \cdot 2}{6}} y^{- \frac{7}{4}}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $4$ by $2$ .

$\frac{2 y^{\frac{1}{3}}}{2 x^{\frac{- 3 + 8}{6}} y^{- \frac{7}{4}}}$

Step 2.7.2

Add $- 3$ and $8$ .

$\frac{2 y^{\frac{1}{3}}}{2 x^{\frac{5}{6}} y^{- \frac{7}{4}}}$

Step 3

Move $y^{- \frac{7}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{2 y^{\frac{1}{3}} y^{\frac{7}{4}}}{2 x^{\frac{5}{6}}}$

Step 4

Multiply $y^{\frac{1}{3}}$ by $y^{\frac{7}{4}}$ by adding the exponents.

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Step 4.1

Move $y^{\frac{7}{4}}$ .

$\frac{2 (y^{\frac{7}{4}} y^{\frac{1}{3}})}{2 x^{\frac{5}{6}}}$

Step 4.2

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

$\frac{2 y^{\frac{7}{4} + \frac{1}{3}}}{2 x^{\frac{5}{6}}}$

Step 4.3

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 y^{\frac{7}{4} \cdot \frac{3}{3} + \frac{1}{3}}}{2 x^{\frac{5}{6}}}$

Step 4.4

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 y^{\frac{7}{4} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{4}{4}}}{2 x^{\frac{5}{6}}}$

Step 4.5

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.5.1

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$\frac{2 y^{\frac{7 \cdot 3}{4 \cdot 3} + \frac{1}{3} \cdot \frac{4}{4}}}{2 x^{\frac{5}{6}}}$

Step 4.5.2

Multiply $4$ by $3$ .

$\frac{2 y^{\frac{7 \cdot 3}{12} + \frac{1}{3} \cdot \frac{4}{4}}}{2 x^{\frac{5}{6}}}$

Step 4.5.3

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{2 y^{\frac{7 \cdot 3}{12} + \frac{4}{3 \cdot 4}}}{2 x^{\frac{5}{6}}}$

Step 4.5.4

Multiply $3$ by $4$ .

$\frac{2 y^{\frac{7 \cdot 3}{12} + \frac{4}{12}}}{2 x^{\frac{5}{6}}}$

Step 4.6

Combine the numerators over the common denominator.

$\frac{2 y^{\frac{7 \cdot 3 + 4}{12}}}{2 x^{\frac{5}{6}}}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $7$ by $3$ .

$\frac{2 y^{\frac{21 + 4}{12}}}{2 x^{\frac{5}{6}}}$

Step 4.7.2

Add $21$ and $4$ .

$\frac{2 y^{\frac{25}{12}}}{2 x^{\frac{5}{6}}}$

Step 5

Cancel the common factor.

$\frac{2 y^{\frac{25}{12}}}{2 x^{\frac{5}{6}}}$

Step 6

Rewrite the expression.

$\frac{y^{\frac{25}{12}}}{x^{\frac{5}{6}}}$