Evaluate 5+( square root of 43+2(3+4^2))÷2

$5 + \sqrt{43 + 2 (3 + 4^{2})} \div 2$

Simplify each term.

Rewrite the division as a fraction.

$5 + \frac{\sqrt{43 + 2 (3 + 4^{2})}}{2}$

Simplify the numerator.

Raise $4$ to the power of $2$ .

$5 + \frac{\sqrt{43 + 2 (3 + 16)}}{2}$

Add $3$ and $16$ .

$5 + \frac{\sqrt{43 + 2 \cdot 19}}{2}$

Multiply $2$ by $19$ .

$5 + \frac{\sqrt{43 + 38}}{2}$

Add $43$ and $38$ .

$5 + \frac{\sqrt{81}}{2}$

Rewrite $81$ as $9^{2}$ .

$5 + \frac{\sqrt{9^{2}}}{2}$

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

$5 + \frac{9}{2}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 \cdot \frac{2}{2} + \frac{9}{2}$

Combine $5$ and $\frac{2}{2}$ .

$\frac{5 \cdot 2}{2} + \frac{9}{2}$

Combine the numerators over the common denominator.

$\frac{5 \cdot 2 + 9}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$\frac{10 + 9}{2}$

Add $10$ and $9$ .

$\frac{19}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{19}{2}$

Decimal Form:

$9.5$

Mixed Number Form:

$9 \frac{1}{2}$