Simplify ( square root of 81+1/2)^2

${(\sqrt{81} + \frac{1}{2})}^{2}$

Simplify each term.

Rewrite

$81$ as

$9^{2}$ .

${(\sqrt{9^{2}} + \frac{1}{2})}^{2}$

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

${(9 + \frac{1}{2})}^{2}$

To write

$9$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

${(9 \cdot \frac{2}{2} + \frac{1}{2})}^{2}$

Combine

$9$ and

$\frac{2}{2}$ .

${(\frac{9 \cdot 2}{2} + \frac{1}{2})}^{2}$

Combine the numerators over the common denominator.

${(\frac{9 \cdot 2 + 1}{2})}^{2}$

Simplify the numerator.

Multiply

$9$ by

$2$ .

${(\frac{18 + 1}{2})}^{2}$

Add

$18$ and

$1$ .

${(\frac{19}{2})}^{2}$

Apply the product rule to

$\frac{19}{2}$ .

$\frac{19^{2}}{2^{2}}$

Raise

$19$ to the power of

$2$ .

$\frac{361}{2^{2}}$

Raise

$2$ to the power of

$2$ .

$\frac{361}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{361}{4}$

Decimal Form:

$90.25$

Mixed Number Form:

$90 \frac{1}{4}$