Precalculus Examples

$\frac{e^{1.5} - e^{- 1.5}}{2}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{e^{1.5} - \frac{1}{e^{1.5}}}{2}$

Step 1.2

To write $e^{1.5}$ as a fraction with a common denominator, multiply by $\frac{e^{1.5}}{e^{1.5}}$ .

$\frac{\frac{e^{1.5} e^{1.5}}{e^{1.5}} - \frac{1}{e^{1.5}}}{2}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{\frac{e^{1.5} e^{1.5} - 1}{e^{1.5}}}{2}$

Step 1.4

Rewrite $\frac{e^{1.5} e^{1.5} - 1}{e^{1.5}}$ in a factored form.

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Step 1.4.1

Multiply $e^{1.5}$ by $e^{1.5}$ by adding the exponents.

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Step 1.4.1.1

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

$\frac{\frac{e^{1.5 + 1.5} - 1}{e^{1.5}}}{2}$

Step 1.4.1.2

Add $1.5$ and $1.5$ .

$\frac{\frac{e^{3} - 1}{e^{1.5}}}{2}$

Step 1.4.2

Rewrite $e^{3} - 1$ in a factored form.

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Step 1.4.2.1

Rewrite $e^{3}$ as ${(e^{1})}^{3}$ .

$\frac{\frac{{(e^{1})}^{3} - 1}{e^{1.5}}}{2}$

Step 1.4.2.2

Rewrite $1$ as $1^{3}$ .

$\frac{\frac{{(e^{1})}^{3} - 1^{3}}{e^{1.5}}}{2}$

Step 1.4.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = e^{1}$ and $b = 1$ .

$\frac{\frac{(e^{1} - 1) ({(e^{1})}^{2} + e^{1} \cdot 1 + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4

Simplify.

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Step 1.4.2.4.1

Simplify.

$\frac{\frac{(e - 1) ({(e^{1})}^{2} + e^{1} \cdot 1 + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4.2

Multiply the exponents in ${(e^{1})}^{2}$ .

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Step 1.4.2.4.2.1

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

$\frac{\frac{(e - 1) (e^{1 \cdot 2} + e^{1} \cdot 1 + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4.2.2

Multiply $2$ by $1$ .

$\frac{\frac{(e - 1) (e^{2} + e^{1} \cdot 1 + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4.3

Simplify.

$\frac{\frac{(e - 1) (e^{2} + e \cdot 1 + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4.4

Multiply $e$ by $1$ .

$\frac{\frac{(e - 1) (e^{2} + e + 1^{2})}{e^{1.5}}}{2}$

Step 1.4.2.4.5

One to any power is one.

$\frac{\frac{(e - 1) (e^{2} + e + 1)}{e^{1.5}}}{2}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{(e - 1) (e^{2} + e + 1)}{e^{1.5}} \cdot \frac{1}{2}$

Step 3

Combine.

$\frac{(e - 1) (e^{2} + e + 1) \cdot 1}{e^{1.5} \cdot 2}$

Step 4

Simplify the expression.

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Step 4.1

Multiply $e - 1$ by $1$ .

$\frac{(e - 1) (e^{2} + e + 1)}{e^{1.5} \cdot 2}$

Step 4.2

Move $2$ to the left of $e^{1.5}$ .

$\frac{(e - 1) (e^{2} + e + 1)}{2 e^{1.5}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{(e - 1) (e^{2} + e + 1)}{2 e^{1.5}}$

Decimal Form:

$2.12927945 \dots$