Precalculus Examples

${(e^{\frac{2}{3}})}^{\ln (\sqrt{27})}$

Step 1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1

Factor $9$ out of $27$ .

${(e^{\frac{2}{3}})}^{\ln (\sqrt{9 (3)})}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

${(e^{\frac{2}{3}})}^{\ln (\sqrt{3^{2} \cdot 3})}$

Step 2

Pull terms out from under the radical.

${(e^{\frac{2}{3}})}^{\ln (3 \sqrt{3})}$

Step 3

Multiply the exponents in ${(e^{\frac{2}{3}})}^{\ln (3 \sqrt{3})}$ .

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Step 3.1

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

$e^{\frac{2}{3} \ln (3 \sqrt{3})}$

Step 3.2

Simplify $\frac{2}{3} \ln (3 \sqrt{3})$ by moving $\frac{2}{3}$ inside the logarithm.

$e^{\ln ({(3 \sqrt{3})}^{\frac{2}{3}})}$

Step 3.3

Apply the product rule to $3 \sqrt{3}$ .

$e^{\ln (3^{\frac{2}{3}} {\sqrt{3}}^{\frac{2}{3}})}$

Step 3.4

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

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Step 3.4.1

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

$e^{\ln (3^{\frac{2}{3}} {(3^{\frac{1}{2}})}^{\frac{2}{3}})}$

Step 3.4.2

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

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{1}{2} \cdot \frac{2}{3}})}$

Step 3.4.3

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{2}{2 \cdot 3}})}$

Step 3.4.4

Multiply $2$ by $3$ .

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{2}{6}})}$

Step 3.4.5

Cancel the common factor of $2$ and $6$ .

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Step 3.4.5.1

Factor $2$ out of $2$ .

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{2 (1)}{6}})}$

Step 3.4.5.2

Cancel the common factors.

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Step 3.4.5.2.1

Factor $2$ out of $6$ .

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{2 \cdot 1}{2 \cdot 3}})}$

Step 3.4.5.2.2

Cancel the common factor.

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{2 \cdot 1}{2 \cdot 3}})}$

Step 3.4.5.2.3

Rewrite the expression.

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}})}$

Step 3.4.6

Rewrite $3^{\frac{1}{3}}$ as $\sqrt[3]{3}$ .

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

Step 3.5

Multiply $3^{\frac{2}{3}} \sqrt[3]{3}$ .

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Step 3.5.1

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

$e^{\ln (3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}})}$

Step 3.5.2

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

$e^{\ln (3^{\frac{2}{3} + \frac{1}{3}})}$

Step 3.5.3

Combine the numerators over the common denominator.

$e^{\ln (3^{\frac{2 + 1}{3}})}$

Step 3.5.4

Add $2$ and $1$ .

$e^{\ln (3^{\frac{3}{3}})}$

Step 3.5.5

Cancel the common factor of $3$ .

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Step 3.5.5.1

Cancel the common factor.

$e^{\ln (3^{\frac{3}{3}})}$

Step 3.5.5.2

Rewrite the expression.

$e^{\ln (3^{1})}$

Step 3.6

Evaluate the exponent.

$e^{\ln (3)}$

Step 4

Exponentiation and log are inverse functions.

$3$