Algebra Examples

$\sqrt[3]{\frac{135}{4}}$

Step 1

Rewrite $\sqrt[3]{\frac{135}{4}}$ as $\frac{\sqrt[3]{135}}{\sqrt[3]{4}}$ .

$\frac{\sqrt[3]{135}}{\sqrt[3]{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $135$ as $3^{3} \cdot 5$ .

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Step 2.1.1

Factor $27$ out of $135$ .

$\frac{\sqrt[3]{27 (5)}}{\sqrt[3]{4}}$

Step 2.1.2

Rewrite $27$ as $3^{3}$ .

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

Step 2.2

Pull terms out from under the radical.

$\frac{3 \sqrt[3]{5}}{\sqrt[3]{4}}$

Step 3

Multiply $\frac{3 \sqrt[3]{5}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{3 \sqrt[3]{5}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 \sqrt[3]{5}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 4.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 4.3

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

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 4.4

Add $1$ and $2$ .

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 4.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 4.5.2

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

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 4.5.3

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

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 4.5.4.2

Rewrite the expression.

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 4.5.5

Evaluate the exponent.

$\frac{3 \sqrt[3]{5} {\sqrt[3]{4}}^{2}}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$\frac{3 \sqrt[3]{5} \sqrt[3]{4^{2}}}{4}$

Step 5.2

Raise $4$ to the power of $2$ .

$\frac{3 \sqrt[3]{5} \sqrt[3]{16}}{4}$

Step 5.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.3.1

Factor $8$ out of $16$ .

$\frac{3 \sqrt[3]{5} \sqrt[3]{8 (2)}}{4}$

Step 5.3.2

Rewrite $8$ as $2^{3}$ .

$\frac{3 \sqrt[3]{5} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 5.4

Pull terms out from under the radical.

$\frac{3 \sqrt[3]{5} \cdot 2 \sqrt[3]{2}}{4}$

Step 5.5

Combine exponents.

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Step 5.5.1

Multiply $2$ by $3$ .

$\frac{6 \sqrt[3]{5} \sqrt[3]{2}}{4}$

Step 5.5.2

Combine using the product rule for radicals.

$\frac{6 \sqrt[3]{2 \cdot 5}}{4}$

Step 5.5.3

Multiply $2$ by $5$ .

$\frac{6 \sqrt[3]{10}}{4}$

Step 6

Cancel the common factor of $6$ and $4$ .

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Step 6.1

Factor $2$ out of $6 \sqrt[3]{10}$ .

$\frac{2 (3 \sqrt[3]{10})}{4}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $4$ .

$\frac{2 (3 \sqrt[3]{10})}{2 (2)}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (3 \sqrt[3]{10})}{2 \cdot 2}$

Step 6.2.3

Rewrite the expression.

$\frac{3 \sqrt[3]{10}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt[3]{10}}{2}$

Decimal Form:

$3.23165203 \dots$