Question

$\sqrt{12 + i16}$
plsss help me fast​

Answer 1

To calculate:

$\sqrt{12+i16} =?$

Let us write concept for value of $i$ first of all.

$1.\ i = \sqrt{-1}\\2.\ i^2=-1$

Taking 4 common inside square root:

$\Rightarrow \sqrt{4(3+i4)}\\\Rightarrow 2\sqrt{(3+i4)} (\because \sqrt4=2) ...... (1)$

Now, let us solve for: $(3+i4)$

Adding and subtracting 1 from $(3+i4)$:

$\Rightarrow (3+i4 +1-1)\\\Rightarrow (3 +1-1 +i4)\\\Rightarrow (4+(-1) +i4) \\\\\text{Using property 2 written above, writing } -1 =i^2\\\Rightarrow (4+i^2 +i4) \\\Rightarrow (2^2+i^2 +2 \times i \times 2)\\\Rightarrow (2+i)^2 (\because a^2+b^2+2 \times a \times b = (a+b)^2)$

Putting value of $(3+i4)$ in equation (1):

$\Rightarrow 2\sqrt{(3+i4)} = 2\sqrt{(2+i)^2}\\\Rightarrow \pm 2(2+i)$

So, the simplified form of $\sqrt{12+i16} =\pm 2(2+i)$