Precalculus Examples

$e^{\frac{1}{2} \cdot \ln (\frac{81}{64})}$

Step 1

Simplify $\frac{1}{2} \ln (\frac{81}{64})$ by moving $\frac{1}{2}$ inside the logarithm.

$e^{\ln ({(\frac{81}{64})}^{\frac{1}{2}})}$

Step 2

Exponentiation and log are inverse functions.

${(\frac{81}{64})}^{\frac{1}{2}}$

Step 3

Apply the product rule to $\frac{81}{64}$ .

$\frac{81^{\frac{1}{2}}}{64^{\frac{1}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $81$ as $9^{2}$ .

$\frac{{(9^{2})}^{\frac{1}{2}}}{64^{\frac{1}{2}}}$

Step 4.2

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

$\frac{9^{2 (\frac{1}{2})}}{64^{\frac{1}{2}}}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{9^{2 (\frac{1}{2})}}{64^{\frac{1}{2}}}$

Step 4.3.2

Rewrite the expression.

$\frac{9^{1}}{64^{\frac{1}{2}}}$

Step 4.4

Evaluate the exponent.

$\frac{9}{64^{\frac{1}{2}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $64$ as $8^{2}$ .

$\frac{9}{{(8^{2})}^{\frac{1}{2}}}$

Step 5.2

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

$\frac{9}{8^{2 (\frac{1}{2})}}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$\frac{9}{8^{2 (\frac{1}{2})}}$

Step 5.3.2

Rewrite the expression.

$\frac{9}{8^{1}}$

Step 5.4

Evaluate the exponent.

$\frac{9}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{8}$

Decimal Form:

$1.125$

Mixed Number Form:

$1 \frac{1}{8}$