Precalculus Examples

$\log (6) \sqrt[5]{\frac{s^{6} t}{36}}$

Step 1

Rewrite $\log (6)$ as $\log (2 (3))$ .

$\log (2 (3)) \sqrt[5]{\frac{s^{6} t}{36}}$

Step 2

Rewrite $\log (2 (3))$ as $\log (2) + \log (3)$ .

$(\log (2) + \log (3)) \sqrt[5]{\frac{s^{6} t}{36}}$

Step 3

Rewrite $\sqrt[5]{\frac{s^{6} t}{36}}$ as $\frac{\sqrt[5]{s^{6} t}}{\sqrt[5]{36}}$ .

$(\log (2) + \log (3)) \frac{\sqrt[5]{s^{6} t}}{\sqrt[5]{36}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $s^{6} t$ as $s^{5} (s t)$ .

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Step 4.1.1

Factor out $s^{5}$ .

$(\log (2) + \log (3)) \frac{\sqrt[5]{s^{5} s t}}{\sqrt[5]{36}}$

Step 4.1.2

Add parentheses.

$(\log (2) + \log (3)) \frac{\sqrt[5]{s^{5} (s t)}}{\sqrt[5]{36}}$

Step 4.2

Pull terms out from under the radical.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t}}{\sqrt[5]{36}}$

Step 5

Multiply $\frac{s \sqrt[5]{s t}}{\sqrt[5]{36}}$ by $\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}}$ .

$(\log (2) + \log (3)) (\frac{s \sqrt[5]{s t}}{\sqrt[5]{36}} \cdot \frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{s \sqrt[5]{s t}}{\sqrt[5]{36}}$ by $\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}}$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{\sqrt[5]{36} {\sqrt[5]{36}}^{4}}$

Step 6.2

Raise $\sqrt[5]{36}$ to the power of $1$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{1} {\sqrt[5]{36}}^{4}}$

Step 6.3

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

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{1 + 4}}$

Step 6.4

Add $1$ and $4$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{5}}$

Step 6.5

Rewrite ${\sqrt[5]{36}}^{5}$ as $36$ .

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Step 6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{36}$ as $36^{\frac{1}{5}}$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{{(36^{\frac{1}{5}})}^{5}}$

Step 6.5.2

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

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{36^{\frac{1}{5} \cdot 5}}$

Step 6.5.3

Combine $\frac{1}{5}$ and $5$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{36^{\frac{5}{5}}}$

Step 6.5.4

Cancel the common factor of $5$ .

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Step 6.5.4.1

Cancel the common factor.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{36^{\frac{5}{5}}}$

Step 6.5.4.2

Rewrite the expression.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{36^{1}}$

Step 6.5.5

Evaluate the exponent.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} {\sqrt[5]{36}}^{4}}{36}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite ${\sqrt[5]{36}}^{4}$ as $\sqrt[5]{36^{4}}$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} \sqrt[5]{36^{4}}}{36}$

Step 7.2

Raise $36$ to the power of $4$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} \sqrt[5]{1679616}}{36}$

Step 7.3

Rewrite $1679616$ as $6^{5} \cdot 216$ .

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Step 7.3.1

Factor $7776$ out of $1679616$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} \sqrt[5]{7776 (216)}}{36}$

Step 7.3.2

Rewrite $7776$ as $6^{5}$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} \sqrt[5]{6^{5} \cdot 216}}{36}$

Step 7.4

Pull terms out from under the radical.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t} \cdot 6 \sqrt[5]{216}}{36}$

Step 7.5

Combine using the product rule for radicals.

$(\log (2) + \log (3)) \frac{s \cdot 6 \sqrt[5]{s t \cdot 216}}{36}$

Step 8

Cancel the common factor of $6$ and $36$ .

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Step 8.1

Factor $6$ out of $s \cdot 6 \sqrt[5]{s t \cdot 216}$ .

$(\log (2) + \log (3)) \frac{6 (s \sqrt[5]{s t \cdot 216})}{36}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $6$ out of $36$ .

$(\log (2) + \log (3)) \frac{6 (s \sqrt[5]{s t \cdot 216})}{6 \cdot 6}$

Step 8.2.2

Cancel the common factor.

$(\log (2) + \log (3)) \frac{6 (s \sqrt[5]{s t \cdot 216})}{6 \cdot 6}$

Step 8.2.3

Rewrite the expression.

$(\log (2) + \log (3)) \frac{s \sqrt[5]{s t \cdot 216}}{6}$

Step 9

Move $216$ to the left of $s t$ .

$(\log (2) + \log (3)) \frac{s \sqrt[5]{216 \cdot (s t)}}{6}$

Step 10

Apply the distributive property.

$\log (2) \frac{s \sqrt[5]{216 s t}}{6} + \log (3) \frac{s \sqrt[5]{216 s t}}{6}$

Step 11

Combine $\log (2)$ and $\frac{s \sqrt[5]{216 s t}}{6}$ .

$\frac{\log (2) (s \sqrt[5]{216 s t})}{6} + \log (3) \frac{s \sqrt[5]{216 s t}}{6}$

Step 12

Combine $\log (3)$ and $\frac{s \sqrt[5]{216 s t}}{6}$ .

$\frac{\log (2) (s \sqrt[5]{216 s t})}{6} + \frac{\log (3) (s \sqrt[5]{216 s t})}{6}$

Step 13

Remove parentheses.

$\frac{\log (2) s \sqrt[5]{216 s t}}{6} + \frac{\log (3) s \sqrt[5]{216 s t}}{6}$

Step 14

Reorder factors in $\frac{\log (2) s \sqrt[5]{216 s t}}{6} + \frac{\log (3) s \sqrt[5]{216 s t}}{6}$ .

$\frac{s \log (2) \sqrt[5]{216 s t}}{6} + \frac{s \log (3) \sqrt[5]{216 s t}}{6}$