Algebra Examples

$\sqrt[3]{24} - 5 \sqrt[3]{81} + \sqrt[3]{\frac{1}{9}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $24$ as $2^{3} \cdot 3$ .

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Step 1.1.1

Factor $8$ out of $24$ .

$\sqrt[3]{8 (3)} - 5 \sqrt[3]{81} + \sqrt[3]{\frac{1}{9}}$

Step 1.1.2

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

$\sqrt[3]{2^{3} \cdot 3} - 5 \sqrt[3]{81} + \sqrt[3]{\frac{1}{9}}$

Step 1.2

Pull terms out from under the radical.

$2 \sqrt[3]{3} - 5 \sqrt[3]{81} + \sqrt[3]{\frac{1}{9}}$

Step 1.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 1.3.1

Factor $27$ out of $81$ .

$2 \sqrt[3]{3} - 5 \sqrt[3]{27 (3)} + \sqrt[3]{\frac{1}{9}}$

Step 1.3.2

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

$2 \sqrt[3]{3} - 5 \sqrt[3]{3^{3} \cdot 3} + \sqrt[3]{\frac{1}{9}}$

Step 1.4

Pull terms out from under the radical.

$2 \sqrt[3]{3} - 5 (3 \sqrt[3]{3}) + \sqrt[3]{\frac{1}{9}}$

Step 1.5

Multiply $3$ by $- 5$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \sqrt[3]{\frac{1}{9}}$

Step 1.6

Rewrite $\sqrt[3]{\frac{1}{9}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{9}}$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{1}}{\sqrt[3]{9}}$

Step 1.7

Any root of $1$ is $1$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{1}{\sqrt[3]{9}}$

Step 1.8

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{1}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 1.9

Combine and simplify the denominator.

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Step 1.9.1

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 1.9.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 1.9.3

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

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 1.9.4

Add $1$ and $2$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 1.9.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 1.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 1.9.5.2

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

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 1.9.5.3

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

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 1.9.5.4

Cancel the common factor of $3$ .

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Step 1.9.5.4.1

Cancel the common factor.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 1.9.5.4.2

Rewrite the expression.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{1}}$

Step 1.9.5.5

Evaluate the exponent.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9}$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{9^{2}}}{9}$

Step 1.10.2

Raise $9$ to the power of $2$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{81}}{9}$

Step 1.10.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 1.10.3.1

Factor $27$ out of $81$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{27 (3)}}{9}$

Step 1.10.3.2

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

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{3^{3} \cdot 3}}{9}$

Step 1.10.4

Pull terms out from under the radical.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{9}$

Step 1.11

Cancel the common factor of $3$ and $9$ .

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Step 1.11.1

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

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{3 (\sqrt[3]{3})}{9}$

Step 1.11.2

Cancel the common factors.

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Step 1.11.2.1

Factor $3$ out of $9$ .

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{3 \cdot 3}$

Step 1.11.2.2

Cancel the common factor.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{3 \cdot 3}$

Step 1.11.2.3

Rewrite the expression.

$2 \sqrt[3]{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{3}}{3}$

Step 2

Find the common denominator.

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Step 2.1

Write $2 \sqrt[3]{3}$ as a fraction with denominator $1$ .

$\frac{2 \sqrt[3]{3}}{1} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{3}}{3}$

Step 2.2

Multiply $\frac{2 \sqrt[3]{3}}{1}$ by $\frac{3}{3}$ .

$\frac{2 \sqrt[3]{3}}{1} \cdot \frac{3}{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{3}}{3}$

Step 2.3

Multiply $\frac{2 \sqrt[3]{3}}{1}$ by $\frac{3}{3}$ .

$\frac{2 \sqrt[3]{3} \cdot 3}{3} - 15 \sqrt[3]{3} + \frac{\sqrt[3]{3}}{3}$

Step 2.4

Write $- 15 \sqrt[3]{3}$ as a fraction with denominator $1$ .

$\frac{2 \sqrt[3]{3} \cdot 3}{3} + \frac{- 15 \sqrt[3]{3}}{1} + \frac{\sqrt[3]{3}}{3}$

Step 2.5

Multiply $\frac{- 15 \sqrt[3]{3}}{1}$ by $\frac{3}{3}$ .

$\frac{2 \sqrt[3]{3} \cdot 3}{3} + \frac{- 15 \sqrt[3]{3}}{1} \cdot \frac{3}{3} + \frac{\sqrt[3]{3}}{3}$

Step 2.6

Multiply $\frac{- 15 \sqrt[3]{3}}{1}$ by $\frac{3}{3}$ .

$\frac{2 \sqrt[3]{3} \cdot 3}{3} + \frac{- 15 \sqrt[3]{3} \cdot 3}{3} + \frac{\sqrt[3]{3}}{3}$

Step 3

Combine the numerators over the common denominator.

$\frac{2 \sqrt[3]{3} \cdot 3 - 15 \sqrt[3]{3} \cdot 3 + \sqrt[3]{3}}{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $2$ .

$\frac{6 \sqrt[3]{3} - 15 \sqrt[3]{3} \cdot 3 + \sqrt[3]{3}}{3}$

Step 4.2

Multiply $3$ by $- 15$ .

$\frac{6 \sqrt[3]{3} - 45 \sqrt[3]{3} + \sqrt[3]{3}}{3}$

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $45 \sqrt[3]{3}$ from $6 \sqrt[3]{3}$ .

$\frac{- 39 \sqrt[3]{3} + \sqrt[3]{3}}{3}$

Step 5.2

Add $- 39 \sqrt[3]{3}$ and $\sqrt[3]{3}$ .

$\frac{- 38 \sqrt[3]{3}}{3}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{38 \sqrt[3]{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{38 \sqrt[3]{3}}{3}$

Decimal Form:

$- 18.26849455 \dots$