Precalculus Examples

$\sqrt{2^{2} + {(- \frac{5}{2})}^{2}}$

Step 1

Raise $2$ to the power of $2$ .

$\sqrt{4 + {(- \frac{5}{2})}^{2}}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.1

Apply the product rule to $- \frac{5}{2}$ .

$\sqrt{4 + {(- 1)}^{2} {(\frac{5}{2})}^{2}}$

Step 2.2

Apply the product rule to $\frac{5}{2}$ .

$\sqrt{4 + {(- 1)}^{2} \frac{5^{2}}{2^{2}}}$

Step 3

Raise $- 1$ to the power of $2$ .

$\sqrt{4 + 1 \frac{5^{2}}{2^{2}}}$

Step 4

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$\sqrt{4 + \frac{5^{2}}{2^{2}}}$

Step 5

Raise $5$ to the power of $2$ .

$\sqrt{4 + \frac{25}{2^{2}}}$

Step 6

Raise $2$ to the power of $2$ .

$\sqrt{4 + \frac{25}{4}}$

Step 7

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{4 \cdot \frac{4}{4} + \frac{25}{4}}$

Step 8

Combine $4$ and $\frac{4}{4}$ .

$\sqrt{\frac{4 \cdot 4}{4} + \frac{25}{4}}$

Step 9

Combine the numerators over the common denominator.

$\sqrt{\frac{4 \cdot 4 + 25}{4}}$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Multiply $4$ by $4$ .

$\sqrt{\frac{16 + 25}{4}}$

Step 10.2

Add $16$ and $25$ .

$\sqrt{\frac{41}{4}}$

Step 11

Rewrite $\sqrt{\frac{41}{4}}$ as $\frac{\sqrt{41}}{\sqrt{4}}$ .

$\frac{\sqrt{41}}{\sqrt{4}}$

Step 12

Simplify the denominator.

Tap for more steps...

Step 12.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{41}}{\sqrt{2^{2}}}$

Step 12.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{41}}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{41}}{2}$

Decimal Form:

$3.20156211 \dots$