Precalculus Examples

$79 + 106 e^{\frac{127}{30} \cdot \ln (\frac{71}{106})}$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{127}{30} \ln (\frac{71}{106})$ by moving $\frac{127}{30}$ inside the logarithm.

$79 + 106 e^{\ln ({(\frac{71}{106})}^{\frac{127}{30}})}$

Step 1.2

Exponentiation and log are inverse functions.

$79 + 106 {(\frac{71}{106})}^{\frac{127}{30}}$

Step 1.3

Apply the product rule to $\frac{71}{106}$ .

$79 + 106 \frac{71^{\frac{127}{30}}}{106^{\frac{127}{30}}}$

Step 1.4

Combine $106$ and $\frac{71^{\frac{127}{30}}}{106^{\frac{127}{30}}}$ .

$79 + \frac{106 \cdot 71^{\frac{127}{30}}}{106^{\frac{127}{30}}}$

Step 1.5

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

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127}{30}} \cdot 106^{- 1}}$

Step 1.6

Multiply $106^{\frac{127}{30}}$ by $106^{- 1}$ by adding the exponents.

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Step 1.6.1

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

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127}{30} - 1}}$

Step 1.6.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127}{30} - 1 \cdot \frac{30}{30}}}$

Step 1.6.3

Combine $- 1$ and $\frac{30}{30}$ .

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127}{30} + \frac{- 1 \cdot 30}{30}}}$

Step 1.6.4

Combine the numerators over the common denominator.

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127 - 1 \cdot 30}{30}}}$

Step 1.6.5

Simplify the numerator.

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Step 1.6.5.1

Multiply $- 1$ by $30$ .

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{127 - 30}{30}}}$

Step 1.6.5.2

Subtract $30$ from $127$ .

$79 + \frac{71^{\frac{127}{30}}}{106^{\frac{97}{30}}}$

Step 2

To write $79$ as a fraction with a common denominator, multiply by $\frac{106^{\frac{97}{30}}}{106^{\frac{97}{30}}}$ .

$79 \cdot \frac{106^{\frac{97}{30}}}{106^{\frac{97}{30}}} + \frac{71^{\frac{127}{30}}}{106^{\frac{97}{30}}}$

Step 3

Combine $79$ and $\frac{106^{\frac{97}{30}}}{106^{\frac{97}{30}}}$ .

$\frac{79 \cdot 106^{\frac{97}{30}}}{106^{\frac{97}{30}}} + \frac{71^{\frac{127}{30}}}{106^{\frac{97}{30}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{79 \cdot 106^{\frac{97}{30}} + 71^{\frac{127}{30}}}{106^{\frac{97}{30}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{79 \cdot 106^{\frac{97}{30}} + 71^{\frac{127}{30}}}{106^{\frac{97}{30}}}$

Decimal Form:

$98.43142773 \dots$