Precalculus Examples

$38 + (72 - 38) \cdot e^{- \frac{1}{2} \cdot \ln (\frac{17}{11}) \cdot 7}$

Step 1

Simplify each term.

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Step 1.1

Subtract $38$ from $72$ .

$38 + 34 \cdot e^{- \frac{1}{2} \cdot \ln (\frac{17}{11}) \cdot 7}$

Step 1.2

Multiply $- \frac{1}{2} \ln (\frac{17}{11})$ .

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Step 1.2.1

Reorder $\ln (\frac{17}{11})$ and $\frac{1}{2}$ .

$38 + 34 \cdot e^{- (\frac{1}{2} \ln (\frac{17}{11})) \cdot 7}$

Step 1.2.2

Simplify $\frac{1}{2} \ln (\frac{17}{11})$ by moving $\frac{1}{2}$ inside the logarithm.

$38 + 34 \cdot e^{- \ln ({(\frac{17}{11})}^{\frac{1}{2}}) \cdot 7}$

Step 1.3

Apply the product rule to $\frac{17}{11}$ .

$38 + 34 \cdot e^{- \ln (\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}}) \cdot 7}$

Step 1.4

Multiply $- \ln (\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}}) \cdot 7$ .

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Step 1.4.1

Multiply $7$ by $- 1$ .

$38 + 34 \cdot e^{- 7 \ln (\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}$

Step 1.4.2

Simplify $- 7 \ln (\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})$ by moving $7$ inside the logarithm.

$38 + 34 \cdot e^{- \ln ({(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7})}$

Step 1.5

Simplify $- \ln ({(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7})$ by moving $- 1$ inside the logarithm.

$38 + 34 \cdot e^{\ln ({({(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7})}^{- 1})}$

Step 1.6

Exponentiation and log are inverse functions.

$38 + 34 \cdot {({(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7})}^{- 1}$

Step 1.7

Multiply the exponents in ${({(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7})}^{- 1}$ .

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Step 1.7.1

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

$38 + 34 \cdot {(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{7 \cdot - 1}$

Step 1.7.2

Multiply $7$ by $- 1$ .

$38 + 34 \cdot {(\frac{17^{\frac{1}{2}}}{11^{\frac{1}{2}}})}^{- 7}$

Step 1.8

Change the sign of the exponent by rewriting the base as its reciprocal.

$38 + 34 \cdot {(\frac{11^{\frac{1}{2}}}{17^{\frac{1}{2}}})}^{7}$

Step 1.9

Apply the product rule to $\frac{11^{\frac{1}{2}}}{17^{\frac{1}{2}}}$ .

$38 + 34 \cdot \frac{{(11^{\frac{1}{2}})}^{7}}{{(17^{\frac{1}{2}})}^{7}}$

Step 1.10

Multiply the exponents in ${(11^{\frac{1}{2}})}^{7}$ .

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Step 1.10.1

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

$38 + 34 \cdot \frac{11^{\frac{1}{2} \cdot 7}}{{(17^{\frac{1}{2}})}^{7}}$

Step 1.10.2

Combine $\frac{1}{2}$ and $7$ .

$38 + 34 \cdot \frac{11^{\frac{7}{2}}}{{(17^{\frac{1}{2}})}^{7}}$

Step 1.11

Multiply the exponents in ${(17^{\frac{1}{2}})}^{7}$ .

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Step 1.11.1

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

$38 + 34 \cdot \frac{11^{\frac{7}{2}}}{17^{\frac{1}{2} \cdot 7}}$

Step 1.11.2

Combine $\frac{1}{2}$ and $7$ .

$38 + 34 \cdot \frac{11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Step 1.12

Combine $34$ and $\frac{11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$ .

$38 + \frac{34 \cdot 11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Step 2

To write $38$ as a fraction with a common denominator, multiply by $\frac{17^{\frac{7}{2}}}{17^{\frac{7}{2}}}$ .

$38 \cdot \frac{17^{\frac{7}{2}}}{17^{\frac{7}{2}}} + \frac{34 \cdot 11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Step 3

Combine $38$ and $\frac{17^{\frac{7}{2}}}{17^{\frac{7}{2}}}$ .

$\frac{38 \cdot 17^{\frac{7}{2}}}{17^{\frac{7}{2}}} + \frac{34 \cdot 11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Step 4

Combine the numerators over the common denominator.

$\frac{38 \cdot 17^{\frac{7}{2}} + 34 \cdot 11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{38 \cdot 17^{\frac{7}{2}} + 34 \cdot 11^{\frac{7}{2}}}{17^{\frac{7}{2}}}$

Decimal Form:

$45.40938377 \dots$