Algebra Examples

$\frac{\sqrt[6]{s^{4} t^{9}}}{\sqrt[3]{s t}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $s^{4} t^{9}$ as $t^{6} (s^{4} t^{3})$ .

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Step 1.1.1

Factor out $t^{6}$ .

$\frac{\sqrt[6]{s^{4} (t^{6} t^{3})}}{\sqrt[3]{s t}}$

Step 1.1.2

Reorder $s^{4}$ and $t^{6}$ .

$\frac{\sqrt[6]{t^{6} s^{4} t^{3}}}{\sqrt[3]{s t}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt[6]{t^{6} (s^{4} t^{3})}}{\sqrt[3]{s t}}$

Step 1.2

Pull terms out from under the radical.

$\frac{t \sqrt[6]{s^{4} t^{3}}}{\sqrt[3]{s t}}$

Step 2

Multiply $\frac{t \sqrt[6]{s^{4} t^{3}}}{\sqrt[3]{s t}}$ by $\frac{{\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{2}}$ .

$\frac{t \sqrt[6]{s^{4} t^{3}}}{\sqrt[3]{s t}} \cdot \frac{{\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{2}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{t \sqrt[6]{s^{4} t^{3}}}{\sqrt[3]{s t}}$ by $\frac{{\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{2}}$ .

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{\sqrt[3]{s t} {\sqrt[3]{s t}}^{2}}$

Step 3.1.2

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

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{1} {\sqrt[3]{s t}}^{2}}$

Step 3.1.3

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

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{1 + 2}}$

Step 3.1.4

Add $1$ and $2$ .

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{\sqrt[3]{s t}}^{3}}$

Step 3.1.5

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

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Step 3.1.5.1

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

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{({(s t)}^{\frac{1}{3}})}^{3}}$

Step 3.1.5.2

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

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{(s t)}^{\frac{1}{3} \cdot 3}}$

Step 3.1.5.3

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

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{(s t)}^{\frac{3}{3}}}$

Step 3.1.5.4

Cancel the common factor of $3$ .

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Step 3.1.5.4.1

Cancel the common factor.

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{(s t)}^{\frac{3}{3}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{{(s t)}^{1}}$

Step 3.1.5.5

Simplify.

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{s t}$

Step 3.2

Cancel the common factor of $t$ .

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Step 3.2.1

Cancel the common factor.

$\frac{t \sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{s t}$

Step 3.2.2

Rewrite the expression.

$\frac{\sqrt[6]{s^{4} t^{3}} {\sqrt[3]{s t}}^{2}}{s}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\sqrt[6]{s^{4} t^{3}} \sqrt[3]{{(s t)}^{2}}}{s}$

Step 4.2

Apply the product rule to $s t$ .

$\frac{\sqrt[6]{s^{4} t^{3}} \sqrt[3]{s^{2} t^{2}}}{s}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite the expression using the least common index of $6$ .

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{s^{2} t^{2}}$ as ${(s^{2} t^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[6]{s^{4} t^{3}} {(s^{2} t^{2})}^{\frac{1}{3}}}{s}$

Step 5.1.2

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

$\frac{\sqrt[6]{s^{4} t^{3}} {(s^{2} t^{2})}^{\frac{2}{6}}}{s}$

Step 5.1.3

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

$\frac{\sqrt[6]{s^{4} t^{3}} \sqrt[6]{{(s^{2} t^{2})}^{2}}}{s}$

Step 5.2

Combine using the product rule for radicals.

$\frac{\sqrt[6]{s^{4} t^{3} {(s^{2} t^{2})}^{2}}}{s}$

Step 5.3

Apply the product rule to $s^{2} t^{2}$ .

$\frac{\sqrt[6]{s^{4} t^{3} {(s^{2})}^{2} {(t^{2})}^{2}}}{s}$

Step 5.4

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

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Step 5.4.1

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

$\frac{\sqrt[6]{s^{4} t^{3} s^{2 \cdot 2} {(t^{2})}^{2}}}{s}$

Step 5.4.2

Multiply $2$ by $2$ .

$\frac{\sqrt[6]{s^{4} t^{3} s^{4} {(t^{2})}^{2}}}{s}$

Step 5.5

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

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Step 5.5.1

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

$\frac{\sqrt[6]{s^{4} t^{3} s^{4} t^{2 \cdot 2}}}{s}$

Step 5.5.2

Multiply $2$ by $2$ .

$\frac{\sqrt[6]{s^{4} t^{3} s^{4} t^{4}}}{s}$

Step 5.6

Combine exponents.

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Step 5.6.1

Multiply $s^{4}$ by $s^{4}$ by adding the exponents.

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Step 5.6.1.1

Move $s^{4}$ .

$\frac{\sqrt[6]{s^{4} s^{4} t^{3} t^{4}}}{s}$

Step 5.6.1.2

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

$\frac{\sqrt[6]{s^{4 + 4} t^{3} t^{4}}}{s}$

Step 5.6.1.3

Add $4$ and $4$ .

$\frac{\sqrt[6]{s^{8} t^{3} t^{4}}}{s}$

Step 5.6.2

Multiply $t^{3}$ by $t^{4}$ by adding the exponents.

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Step 5.6.2.1

Move $t^{4}$ .

$\frac{\sqrt[6]{s^{8} (t^{4} t^{3})}}{s}$

Step 5.6.2.2

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

$\frac{\sqrt[6]{s^{8} t^{4 + 3}}}{s}$

Step 5.6.2.3

Add $4$ and $3$ .

$\frac{\sqrt[6]{s^{8} t^{7}}}{s}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $s^{8} t^{7}$ as ${(s t)}^{6} (s^{2} t)$ .

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Step 6.1.1

Factor out $s^{6}$ .

$\frac{\sqrt[6]{s^{6} s^{2} t^{7}}}{s}$

Step 6.1.2

Factor out $t^{6}$ .

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

Step 6.1.3

Move $s^{2}$ .

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

Step 6.1.4

Rewrite $s^{6} t^{6}$ as ${(s t)}^{6}$ .

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

Step 6.1.5

Add parentheses.

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

Step 6.2

Pull terms out from under the radical.

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

Step 7

Cancel the common factor of $s$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $t \sqrt[6]{s^{2} t}$ by $1$ .

$t \sqrt[6]{s^{2} t}$