Precalculus Examples

$\log (5) (\frac{25 x^{2}}{\sqrt[3]{5}})$

Step 1

Multiply $\frac{25 x^{2}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\log (5) (\frac{25 x^{2}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}})$

Step 2

Combine and simplify the denominator.

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Multiply $\frac{25 x^{2}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

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

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Add $1$ and $2$ .

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

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

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

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

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Rewrite the expression.

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{5^{1}}$

Evaluate the exponent.

$\log (5) \frac{25 x^{2} {\sqrt[3]{5}}^{2}}{5}$

Step 3

Cancel the common factor of $25$ and $5$ .

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Factor $5$ out of $25 x^{2} {\sqrt[3]{5}}^{2}$ .

$\log (5) \frac{5 (5 x^{2} {\sqrt[3]{5}}^{2})}{5}$

Cancel the common factors.

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Factor $5$ out of $5$ .

$\log (5) \frac{5 (5 x^{2} {\sqrt[3]{5}}^{2})}{5 (1)}$

Cancel the common factor.

$\log (5) \frac{5 (5 x^{2} {\sqrt[3]{5}}^{2})}{5 \cdot 1}$

Rewrite the expression.

$\log (5) \frac{5 x^{2} {\sqrt[3]{5}}^{2}}{1}$

Divide $5 x^{2} {\sqrt[3]{5}}^{2}$ by $1$ .

$\log (5) (5 x^{2} {\sqrt[3]{5}}^{2})$

Step 4

Rewrite ${\sqrt[3]{5}}^{2}$ as ${(5^{2})}^{\frac{1}{3}}$ .

$\log (5) (5 x^{2} \sqrt[3]{5^{2}})$

Step 5

Raise $5$ to the power of $2$ .

$\log (5) (5 x^{2} \sqrt[3]{25})$

Step 6

Move $5$ to the left of $\log (5)$ .

$5 \cdot \log (5) x^{2} \sqrt[3]{25}$

Step 7

Reorder factors in $5 \log (5) x^{2} \sqrt[3]{25}$ .

$5 x^{2} \log (5) \sqrt[3]{25}$