Finite Math Examples

$\sqrt{e^{15 t}} = e^{13}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e^{15 t}}$ as $e^{\frac{15 t}{2}}$ .

$e^{\frac{15 t}{2}} = e^{13}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$\frac{15 t}{2} = 13$

Step 3

Solve for $t$ .

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Step 3.1

Multiply both sides of the equation by $\frac{2}{15}$ .

$\frac{2}{15} \cdot \frac{15 t}{2} = \frac{2}{15} \cdot 13$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{2}{15} \cdot \frac{15 t}{2}$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{2}{15} \cdot \frac{15 t}{2} = \frac{2}{15} \cdot 13$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{15} (15 t) = \frac{2}{15} \cdot 13$

Step 3.2.1.1.2

Cancel the common factor of $15$ .

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Step 3.2.1.1.2.1

Factor $15$ out of $15 t$ .

$\frac{1}{15} (15 (t)) = \frac{2}{15} \cdot 13$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{15} (15 t) = \frac{2}{15} \cdot 13$

Step 3.2.1.1.2.3

Rewrite the expression.

$t = \frac{2}{15} \cdot 13$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $\frac{2}{15} \cdot 13$ .

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Step 3.2.2.1.1

Combine $\frac{2}{15}$ and $13$ .

$t = \frac{2 \cdot 13}{15}$

Step 3.2.2.1.2

Multiply $2$ by $13$ .

$t = \frac{26}{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$t = \frac{26}{15}$

Decimal Form:

$t = 1.7\overline{3}$

Mixed Number Form:

$t = 1 \frac{11}{15}$