Algebra Examples

$\sqrt[6]{\frac{t^{- 8}}{t^{- 4}}}$

Step 1

Reduce the expression $\frac{t^{- 8}}{t^{- 4}}$ by cancelling the common factors.

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Step 1.1

Multiply by $1$ .

$\sqrt[6]{\frac{t^{- 8} \cdot 1}{t^{- 4}}}$

Step 1.2

Factor $t^{- 8}$ out of $t^{- 4}$ .

$\sqrt[6]{\frac{t^{- 8} \cdot 1}{t^{- 8} t^{4}}}$

Step 1.3

Cancel the common factor.

$\sqrt[6]{\frac{t^{- 8} \cdot 1}{t^{- 8} t^{4}}}$

Step 1.4

Rewrite the expression.

$\sqrt[6]{\frac{1}{t^{4}}}$

Step 2

Simplify the expression.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\sqrt[6]{\frac{1^{2}}{t^{4}}}$

Step 2.2

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\sqrt[6]{\frac{1^{2}}{{(t^{2})}^{2}}}$

Step 3

Rewrite $\frac{1^{2}}{{(t^{2})}^{2}}$ as ${(\frac{1}{t^{2}})}^{2}$ .

$\sqrt[6]{{(\frac{1}{t^{2}})}^{2}}$

Step 4

Rewrite $\sqrt[6]{{(\frac{1}{t^{2}})}^{2}}$ as $\sqrt[3]{\sqrt{{(\frac{1}{t^{2}})}^{2}}}$ .

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

Step 5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

Rewrite $\sqrt[3]{\frac{1}{t^{2}}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{t^{2}}}$ .

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

Step 7

Any root of $1$ is $1$ .

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

$\frac{{\sqrt[3]{t^{2}}}^{2}}{\sqrt[3]{t^{2}} {\sqrt[3]{t^{2}}}^{2}}$

Step 9.2

Raise $\sqrt[3]{t^{2}}$ to the power of $1$ .

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

Step 9.3

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

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

Step 9.4

Add $1$ and $2$ .

$\frac{{\sqrt[3]{t^{2}}}^{2}}{{\sqrt[3]{t^{2}}}^{3}}$

Step 9.5

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

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Step 9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t^{2}}$ as $t^{\frac{2}{3}}$ .

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

Step 9.5.2

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

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

Step 9.5.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 9.5.4

Multiply $2$ by $3$ .

$\frac{{\sqrt[3]{t^{2}}}^{2}}{t^{\frac{6}{3}}}$

Step 9.5.5

Cancel the common factor of $6$ and $3$ .

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Step 9.5.5.1

Factor $3$ out of $6$ .

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

Step 9.5.5.2

Cancel the common factors.

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Step 9.5.5.2.1

Factor $3$ out of $3$ .

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

Step 9.5.5.2.2

Cancel the common factor.

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

Step 9.5.5.2.3

Rewrite the expression.

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

Step 9.5.5.2.4

Divide $2$ by $1$ .

$\frac{{\sqrt[3]{t^{2}}}^{2}}{t^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite ${\sqrt[3]{t^{2}}}^{2}$ as $\sqrt[3]{{(t^{2})}^{2}}$ .

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

Step 10.2

Multiply the exponents in ${(t^{2})}^{2}$ .

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Step 10.2.1

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

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

Step 10.2.2

Multiply $2$ by $2$ .

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

Step 10.3

Factor out $t^{3}$ .

$\frac{\sqrt[3]{t^{3} t}}{t^{2}}$

Step 10.4

Pull terms out from under the radical.

$\frac{t \sqrt[3]{t}}{t^{2}}$

Step 11

Cancel the common factor of $t$ and $t^{2}$ .

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Step 11.1

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

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $t$ out of $t^{2}$ .

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

Step 11.2.2

Cancel the common factor.

$\frac{t \sqrt[3]{t}}{t \cdot t}$

Step 11.2.3

Rewrite the expression.

$\frac{\sqrt[3]{t}}{t}$