Question

sqrt{(-4)^{2}+(12)^{2}}

Explanation:

Step1: Assume the expression is $\sqrt{(-4)^2 + (12)^2}$ (interpreting the handwritten as a square - root of sum of squares)

First, calculate the squares of the numbers inside the square - root.
For $(-4)^2$, by the rule of squaring a negative number $(-a)^2=a^2$, so $(-4)^2 = 16$.
For $12^2$, we know that $12\times12 = 144$.

Step2: Calculate the sum inside the square - root

Add the two squared values: $16 + 144=160$.

Step3: Simplify the square - root

We can factorize $160$ as $16\times10$. Then $\sqrt{160}=\sqrt{16\times10}$.
By the property of square - roots $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a = 16$ and $b = 10$ and $a\geq0,b\geq0$), we have $\sqrt{16\times10}=\sqrt{16}\times\sqrt{10}$.
Since $\sqrt{16} = 4$, then $\sqrt{160}=4\sqrt{10}\approx4\times3.162 = 12.648$. If we consider the original expression might be a simple arithmetic (maybe a typo and the square - root is not there, and it's $(-4)^2+(12)^2$), then $16 + 144 = 160$. If it's a multiplication of squares like $(-4)^2\times(12)^2$, then $(16)\times(144)=2304$ and $\sqrt{2304}=48$. Let's assume the most probable is the square - root of sum of squares as a common Pythagorean - like expression. So $\sqrt{(-4)^2+(12)^2}=\sqrt{16 + 144}=\sqrt{160}=4\sqrt{10}\approx12.65$ or if it's $(-4)^2\times(12)^2=16\times144 = 2304$ and $\sqrt{2304}=48$. If we take the expression as $(-4)^2+(12)^2$ (no square - root), the result is $160$.

If we assume the expression is $\sqrt{(-4)^{2}\times(12)^{2}}$:

Step1: Calculate the squares

$(-4)^2 = 16$ and $12^2=144$.

Step2: Multiply the squares

$16\times144 = 2304$.

Step3: Take the square - root

$\sqrt{2304}=48$ (since $48\times48 = 2304$)

Answer:

If the expression is $\sqrt{(-4)^2+(12)^2}$, the answer is $4\sqrt{10}\approx12.65$ (or $160$ without square - root, or $48$ if it's $\sqrt{(-4)^2\times(12)^2}$). The most probable (if it's a distance formula - like) is $4\sqrt{10}\approx12.65$ or $48$ (if it's product of squares under root). If we consider the expression as $(-4)^2+(12)^2$, the answer is $160$.