Precalculus Examples

$79 + 106 e^{(\frac{45}{30}) \ln (\frac{79}{106})}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $45$ and $30$ .

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Step 1.1.1

Factor $15$ out of $45$ .

$79 + 106 e^{\frac{15 (3)}{30} \ln (\frac{79}{106})}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $15$ out of $30$ .

$79 + 106 e^{\frac{15 \cdot 3}{15 \cdot 2} \ln (\frac{79}{106})}$

Step 1.1.2.2

Cancel the common factor.

$79 + 106 e^{\frac{15 \cdot 3}{15 \cdot 2} \ln (\frac{79}{106})}$

Step 1.1.2.3

Rewrite the expression.

$79 + 106 e^{\frac{3}{2} \ln (\frac{79}{106})}$

Step 1.2

Simplify $\frac{3}{2} \ln (\frac{79}{106})$ by moving $\frac{3}{2}$ inside the logarithm.

$79 + 106 e^{\ln ({(\frac{79}{106})}^{\frac{3}{2}})}$

Step 1.3

Exponentiation and log are inverse functions.

$79 + 106 {(\frac{79}{106})}^{\frac{3}{2}}$

Step 1.4

Apply the product rule to $\frac{79}{106}$ .

$79 + 106 \frac{79^{\frac{3}{2}}}{106^{\frac{3}{2}}}$

Step 1.5

Combine $106$ and $\frac{79^{\frac{3}{2}}}{106^{\frac{3}{2}}}$ .

$79 + \frac{106 \cdot 79^{\frac{3}{2}}}{106^{\frac{3}{2}}}$

Step 1.6

Move $106^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3}{2}} \cdot 106^{- 1}}$

Step 1.7

Multiply $106^{\frac{3}{2}}$ by $106^{- 1}$ by adding the exponents.

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Step 1.7.1

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

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3}{2} - 1}}$

Step 1.7.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3}{2} - 1 \cdot \frac{2}{2}}}$

Step 1.7.3

Combine $- 1$ and $\frac{2}{2}$ .

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}}$

Step 1.7.4

Combine the numerators over the common denominator.

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3 - 1 \cdot 2}{2}}}$

Step 1.7.5

Simplify the numerator.

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Step 1.7.5.1

Multiply $- 1$ by $2$ .

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{3 - 2}{2}}}$

Step 1.7.5.2

Subtract $2$ from $3$ .

$79 + \frac{79^{\frac{3}{2}}}{106^{\frac{1}{2}}}$

Step 2

To write $79$ as a fraction with a common denominator, multiply by $\frac{106^{\frac{1}{2}}}{106^{\frac{1}{2}}}$ .

$79 \cdot \frac{106^{\frac{1}{2}}}{106^{\frac{1}{2}}} + \frac{79^{\frac{3}{2}}}{106^{\frac{1}{2}}}$

Step 3

Combine $79$ and $\frac{106^{\frac{1}{2}}}{106^{\frac{1}{2}}}$ .

$\frac{79 \cdot 106^{\frac{1}{2}}}{106^{\frac{1}{2}}} + \frac{79^{\frac{3}{2}}}{106^{\frac{1}{2}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{79 \cdot 106^{\frac{1}{2}} + 79^{\frac{3}{2}}}{106^{\frac{1}{2}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{79 \cdot 106^{\frac{1}{2}} + 79^{\frac{3}{2}}}{106^{\frac{1}{2}}}$

Decimal Form:

$147.20052287 \dots$