Algebra Examples

$\frac{\sqrt[3]{a b}}{\sqrt[3]{25}}$

Step 1

Multiply $\frac{\sqrt[3]{a b}}{\sqrt[3]{25}}$ by $\frac{{\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{2}}$ .

$\frac{\sqrt[3]{a b}}{\sqrt[3]{25}} \cdot \frac{{\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[3]{a b}}{\sqrt[3]{25}}$ by $\frac{{\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{2}}$ .

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{\sqrt[3]{25} {\sqrt[3]{25}}^{2}}$

Step 2.2

Raise $\sqrt[3]{25}$ to the power of $1$ .

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{1} {\sqrt[3]{25}}^{2}}$

Step 2.3

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

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{{\sqrt[3]{25}}^{3}}$

Step 2.5

Rewrite ${\sqrt[3]{25}}^{3}$ as $25$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{{(25^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{25^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

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

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{25^{\frac{3}{3}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{25^{\frac{3}{3}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{25^{1}}$

Step 2.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{a b} {\sqrt[3]{25}}^{2}}{25}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{25}}^{2}$ as $\sqrt[3]{25^{2}}$ .

$\frac{\sqrt[3]{a b} \sqrt[3]{25^{2}}}{25}$

Step 3.2

Raise $25$ to the power of $2$ .

$\frac{\sqrt[3]{a b} \sqrt[3]{625}}{25}$

Step 3.3

Rewrite $625$ as $5^{3} \cdot 5$ .

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Step 3.3.1

Factor $125$ out of $625$ .

$\frac{\sqrt[3]{a b} \sqrt[3]{125 (5)}}{25}$

Step 3.3.2

Rewrite $125$ as $5^{3}$ .

$\frac{\sqrt[3]{a b} \sqrt[3]{5^{3} \cdot 5}}{25}$

Step 3.4

Pull terms out from under the radical.

$\frac{\sqrt[3]{a b} \cdot 5 \sqrt[3]{5}}{25}$

Step 3.5

Combine using the product rule for radicals.

$\frac{5 \sqrt[3]{a b \cdot 5}}{25}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $5$ and $25$ .

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Step 4.1.1

Factor $5$ out of $5 \sqrt[3]{a b \cdot 5}$ .

$\frac{5 (\sqrt[3]{a b \cdot 5})}{25}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $5$ out of $25$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

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

Step 4.2

Reorder factors in $\frac{\sqrt[3]{a b \cdot 5}}{5}$ .

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