Algebra Examples

$\frac{a^{\frac{2}{3}}}{b^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}}}{a^{\frac{1}{3}}}$

Step 1

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

$\frac{a^{\frac{2}{3}}}{b^{\frac{1}{3}}} \cdot \frac{a^{\frac{1}{3}}}{a^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}}}{a^{\frac{1}{3}}}$

Step 2

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

$\frac{a^{\frac{2}{3}}}{b^{\frac{1}{3}}} \cdot \frac{a^{\frac{1}{3}}}{a^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}}}{a^{\frac{1}{3}}} \cdot \frac{b^{\frac{1}{3}}}{b^{\frac{1}{3}}}$

Step 3

Write each expression with a common denominator of $b^{\frac{1}{3}} a^{\frac{1}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{a^{\frac{2}{3}}}{b^{\frac{1}{3}}}$ by $\frac{a^{\frac{1}{3}}}{a^{\frac{1}{3}}}$ .

$\frac{a^{\frac{2}{3}} a^{\frac{1}{3}}}{b^{\frac{1}{3}} a^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}}}{a^{\frac{1}{3}}} \cdot \frac{b^{\frac{1}{3}}}{b^{\frac{1}{3}}}$

Step 3.2

Multiply $\frac{b^{\frac{2}{3}}}{a^{\frac{1}{3}}}$ by $\frac{b^{\frac{1}{3}}}{b^{\frac{1}{3}}}$ .

$\frac{a^{\frac{2}{3}} a^{\frac{1}{3}}}{b^{\frac{1}{3}} a^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 3.3

Reorder the factors of $b^{\frac{1}{3}} a^{\frac{1}{3}}$ .

$\frac{a^{\frac{2}{3}} a^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}} + \frac{b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{a^{\frac{2}{3}} a^{\frac{1}{3}} + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{a^{\frac{2}{3} + \frac{1}{3}} + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.1.2

Combine the numerators over the common denominator.

$\frac{a^{\frac{2 + 1}{3}} + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.1.3

Add $2$ and $1$ .

$\frac{a^{\frac{3}{3}} + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.1.4

Divide $3$ by $3$ .

$\frac{a^{1} + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.2

Simplify $a^{1}$ .

$\frac{a + b^{\frac{2}{3}} b^{\frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.3

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

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Step 5.3.1

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

$\frac{a + b^{\frac{2}{3} + \frac{1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.3.2

Combine the numerators over the common denominator.

$\frac{a + b^{\frac{2 + 1}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.3.3

Add $2$ and $1$ .

$\frac{a + b^{\frac{3}{3}}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.3.4

Divide $3$ by $3$ .

$\frac{a + b^{1}}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$

Step 5.4

Simplify $b^{1}$ .

$\frac{a + b}{a^{\frac{1}{3}} b^{\frac{1}{3}}}$