Calculus Examples

$\frac{1}{1 + \frac{1}{2}} \cdot 3^{1 + \frac{1}{2}}$

Step 1

Simplify the denominator.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\frac{1}{\frac{2}{2} + \frac{1}{2}} \cdot 3^{1 + \frac{1}{2}}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{1}{\frac{2 + 1}{2}} \cdot 3^{1 + \frac{1}{2}}$

Step 1.3

Add $2$ and $1$ .

$\frac{1}{\frac{3}{2}} \cdot 3^{1 + \frac{1}{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{2}{3}) \cdot 3^{1 + \frac{1}{2}}$

Step 3

Simplify the expression.

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Step 3.1

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} \cdot 3^{1 + \frac{1}{2}}$

Step 3.2

Write $1$ as a fraction with a common denominator.

$\frac{2}{3} \cdot 3^{\frac{2}{2} + \frac{1}{2}}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{2}{3} \cdot 3^{\frac{2 + 1}{2}}$

Step 3.4

Add $2$ and $1$ .

$\frac{2}{3} \cdot 3^{\frac{3}{2}}$

Step 4

Combine $\frac{2}{3}$ and $3^{\frac{3}{2}}$ .

$\frac{2 \cdot 3^{\frac{3}{2}}}{3}$

Step 5

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2 \cdot 3^{\frac{3}{2}} \cdot 3^{- 1}$

Step 6

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

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Step 6.1

Move $3^{- 1}$ .

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

Step 6.2

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

$2 \cdot 3^{- 1 + \frac{3}{2}}$

Step 6.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \cdot 3^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}$

Step 6.4

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

$2 \cdot 3^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}$

Step 6.5

Combine the numerators over the common denominator.

$2 \cdot 3^{\frac{- 1 \cdot 2 + 3}{2}}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $- 1$ by $2$ .

$2 \cdot 3^{\frac{- 2 + 3}{2}}$

Step 6.6.2

Add $- 2$ and $3$ .

$2 \cdot 3^{\frac{1}{2}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 \cdot 3^{\frac{1}{2}}$

Decimal Form:

$3.46410161 \dots$