Precalculus Examples

$\sqrt{4 s t^{5}} \sqrt[8]{s^{4} t^{7}}$

Step 1

Rewrite $4 s t^{5}$ as ${(2 t^{2})}^{2} (s t)$ .

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} s t^{5}} \sqrt[8]{s^{4} t^{7}}$

Step 1.2

Factor out $t^{4}$ .

$\sqrt{2^{2} s (t^{4} t)} \sqrt[8]{s^{4} t^{7}}$

Step 1.3

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

$\sqrt{2^{2} s ({(t^{2})}^{2} t)} \sqrt[8]{s^{4} t^{7}}$

Step 1.4

Move $s$ .

$\sqrt{2^{2} {(t^{2})}^{2} s t} \sqrt[8]{s^{4} t^{7}}$

Step 1.5

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

$\sqrt{{(2 t^{2})}^{2} s t} \sqrt[8]{s^{4} t^{7}}$

Step 1.6

Add parentheses.

$\sqrt{{(2 t^{2})}^{2} (s t)} \sqrt[8]{s^{4} t^{7}}$

Step 2

Pull terms out from under the radical.

$2 t^{2} \sqrt{s t} \sqrt[8]{s^{4} t^{7}}$

Step 3

Multiply $2 t^{2} \sqrt{s t} \sqrt[8]{s^{4} t^{7}}$ .

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Step 3.1

Rewrite the expression using the least common index of $8$ .

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Step 3.1.1

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

$2 t^{2} (\sqrt[8]{s^{4} t^{7}} {(s t)}^{\frac{1}{2}})$

Step 3.1.2

Rewrite ${(s t)}^{\frac{1}{2}}$ as ${(s t)}^{\frac{4}{8}}$ .

$2 t^{2} (\sqrt[8]{s^{4} t^{7}} {(s t)}^{\frac{4}{8}})$

Step 3.1.3

Rewrite ${(s t)}^{\frac{4}{8}}$ as $\sqrt[8]{{(s t)}^{4}}$ .

$2 t^{2} (\sqrt[8]{s^{4} t^{7}} \sqrt[8]{{(s t)}^{4}})$

Step 3.2

Combine using the product rule for radicals.

$2 t^{2} \sqrt[8]{s^{4} t^{7} {(s t)}^{4}}$

Step 3.3

Apply the product rule to $s t$ .

$2 t^{2} \sqrt[8]{s^{4} t^{7} s^{4} t^{4}}$

Step 3.4

Combine exponents.

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Step 3.4.1

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

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Step 3.4.1.1

Move $s^{4}$ .

$2 t^{2} \sqrt[8]{s^{4} s^{4} t^{7} t^{4}}$

Step 3.4.1.2

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

$2 t^{2} \sqrt[8]{s^{4 + 4} t^{7} t^{4}}$

Step 3.4.1.3

Add $4$ and $4$ .

$2 t^{2} \sqrt[8]{s^{8} t^{7} t^{4}}$

Step 3.4.2

Multiply $t^{7}$ by $t^{4}$ by adding the exponents.

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Step 3.4.2.1

Move $t^{4}$ .

$2 t^{2} \sqrt[8]{s^{8} (t^{4} t^{7})}$

Step 3.4.2.2

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

$2 t^{2} \sqrt[8]{s^{8} t^{4 + 7}}$

Step 3.4.2.3

Add $4$ and $7$ .

$2 t^{2} \sqrt[8]{s^{8} t^{11}}$

Step 4

Rewrite $s^{8} t^{11}$ as ${(s t)}^{8} t^{3}$ .

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Step 4.1

Factor out $t^{8}$ .

$2 t^{2} \sqrt[8]{s^{8} (t^{8} t^{3})}$

Step 4.2

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

$2 t^{2} \sqrt[8]{{(s t)}^{8} t^{3}}$

Step 5

Pull terms out from under the radical.

$2 t^{2} (s t \sqrt[8]{t^{3}})$

Step 6

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 6.1

Move $t$ .

$2 (t \cdot t^{2}) (s \sqrt[8]{t^{3}})$

Step 6.2

Multiply $t$ by $t^{2}$ .

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Step 6.2.1

Raise $t$ to the power of $1$ .

$2 (t^{1} t^{2}) (s \sqrt[8]{t^{3}})$

Step 6.2.2

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

$2 t^{1 + 2} (s \sqrt[8]{t^{3}})$

Step 6.3

Add $1$ and $2$ .

$2 t^{3} (s \sqrt[8]{t^{3}})$

Step 7

Reorder factors in $2 t^{3} (s \sqrt[8]{t^{3}})$ .

$2 t^{3} s \sqrt[8]{t^{3}}$