Finite Math Examples

$\sqrt[3]{a^{4} b^{5}} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 1

Rewrite $a^{4} b^{5}$ as ${(a b)}^{3} (a b^{2})$ .

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Step 1.1

Factor out $a^{3}$ .

$\sqrt[3]{a^{3} a b^{5}} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 1.2

Factor out $b^{3}$ .

$\sqrt[3]{a^{3} a (b^{3} b^{2})} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 1.3

Move $a$ .

$\sqrt[3]{a^{3} b^{3} a b^{2}} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 1.4

Rewrite $a^{3} b^{3}$ as ${(a b)}^{3}$ .

$\sqrt[3]{{(a b)}^{3} a b^{2}} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 1.5

Add parentheses.

$\sqrt[3]{{(a b)}^{3} (a b^{2})} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 2

Pull terms out from under the radical.

$a b \sqrt[3]{a b^{2}} \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 3

Cancel the common factor of $a$ .

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Step 3.1

Factor $a$ out of $a b \sqrt[3]{a b^{2}}$ .

$a (b \sqrt[3]{a b^{2}}) \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 3.2

Cancel the common factor.

$a (b \sqrt[3]{a b^{2}}) \cdot \frac{b^{\frac{1}{3}}}{a}$

Step 3.3

Rewrite the expression.

$b \sqrt[3]{a b^{2}} \cdot b^{\frac{1}{3}}$

Step 4

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

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Step 4.1

Move $b^{\frac{1}{3}}$ .

$b^{\frac{1}{3}} b \sqrt[3]{a b^{2}}$

Step 4.2

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

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Step 4.2.1

Raise $b$ to the power of $1$ .

$b^{\frac{1}{3}} b^{1} \sqrt[3]{a b^{2}}$

Step 4.2.2

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

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

Step 4.3

Write $1$ as a fraction with a common denominator.

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Add $1$ and $3$ .

$b^{\frac{4}{3}} \sqrt[3]{a b^{2}}$