Precalculus Examples

$\frac{\log_{3} (3 \sqrt{3})}{\sqrt[3]{3}}$

Step 1

Multiply $\frac{\log_{3} (3 \sqrt{3})}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\log_{3} (3 \sqrt{3})}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Evaluate the exponent.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Raise $3$ to the power of $2$ .

$\frac{\log_{3} (3 \sqrt{3}) \sqrt[3]{9}}{3}$

Step 4

Rewrite $\log_{3} (3 \sqrt{3})$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \sqrt[3]{9}}{3}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (3 \sqrt{3})}{\log (3)} \sqrt[3]{9}}{3}$

Step 5

Combine $\frac{\log (3 \sqrt{3})}{\log (3)}$ and $\sqrt[3]{9}$ .

$\frac{\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3)}}{3}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3)} \cdot \frac{1}{3}$

Step 7

Multiply $\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3)} \cdot \frac{1}{3}$ .

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Step 7.1

Multiply $\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3)}$ by $\frac{1}{3}$ .

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3) \cdot 3}$

Step 7.2

Reorder $\log (3)$ and $3$ .

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{3 \cdot \log (3)}$

Step 7.3

Simplify $3 \cdot \log (3)$ by moving $3$ inside the logarithm.

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (3^{3})}$

Step 8

Raise $3$ to the power of $3$ .

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (27)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (3 \sqrt{3}) \sqrt[3]{9}}{\log (27)}$

Decimal Form:

$1.04004191 \dots$