Precalculus Examples

$2000 e^{\frac{1}{2} \cdot \ln (\frac{19}{10}) \cdot 5}$

Step 1

Simplify $\frac{1}{2} \ln (\frac{19}{10})$ by moving $\frac{1}{2}$ inside the logarithm.

$2000 e^{\ln ({(\frac{19}{10})}^{\frac{1}{2}}) \cdot 5}$

Step 2

Apply the product rule to $\frac{19}{10}$ .

$2000 e^{\ln (\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}}) \cdot 5}$

Step 3

Multiply $\ln (\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}}) \cdot 5$ .

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Step 3.1

Reorder $\ln (\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}})$ and $5$ .

$2000 e^{5 \cdot \ln (\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}})}$

Step 3.2

Simplify $5 \cdot \ln (\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}})$ by moving $5$ inside the logarithm.

$2000 e^{\ln ({(\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}})}^{5})}$

Step 4

Exponentiation and log are inverse functions.

$2000 {(\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}})}^{5}$

Step 5

Apply basic rules of exponents.

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Step 5.1

Apply the product rule to $\frac{19^{\frac{1}{2}}}{10^{\frac{1}{2}}}$ .

$2000 \frac{{(19^{\frac{1}{2}})}^{5}}{{(10^{\frac{1}{2}})}^{5}}$

Step 5.2

Multiply the exponents in ${(19^{\frac{1}{2}})}^{5}$ .

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Step 5.2.1

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

$2000 \frac{19^{\frac{1}{2} \cdot 5}}{{(10^{\frac{1}{2}})}^{5}}$

Step 5.2.2

Combine $\frac{1}{2}$ and $5$ .

$2000 \frac{19^{\frac{5}{2}}}{{(10^{\frac{1}{2}})}^{5}}$

Step 5.3

Multiply the exponents in ${(10^{\frac{1}{2}})}^{5}$ .

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Step 5.3.1

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

$2000 \frac{19^{\frac{5}{2}}}{10^{\frac{1}{2} \cdot 5}}$

Step 5.3.2

Combine $\frac{1}{2}$ and $5$ .

$2000 \frac{19^{\frac{5}{2}}}{10^{\frac{5}{2}}}$

Step 6

Combine $2000$ and $\frac{19^{\frac{5}{2}}}{10^{\frac{5}{2}}}$ .

$\frac{2000 \cdot 19^{\frac{5}{2}}}{10^{\frac{5}{2}}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2000 \cdot 19^{\frac{5}{2}}}{10^{\frac{5}{2}}}$

Decimal Form:

$9952.08319900 \dots$