Solve for the Missing Square Root: 7¹ + 3⁴ = 4³ + √x + 2³

First, let's calculate by direct computation the values of all terms in the equation:

$7^1+3^4=4^3+\sqrt{\textcolor{red}{☐}}+2^3 \\ 7+81=64+\sqrt{\textcolor{red}{☐}}+8$

remembering that raising any number to the power of 1 will always give the number itself,

Next, we'll remember that this is an equation and therefore we can move terms from one side to the other, doing so while remembering that when a term moves sides it changes its sign:

$7+81=64+\sqrt{\textcolor{red}{☐}}+8 \\ 7+81-64-8=\sqrt{\textcolor{red}{☐}}\\ 16 =\sqrt{\textcolor{red}{☐}}$In the final step, we simplified the left side of the equation by combining like terms,

Now let's examine the equation we got:

On the left side we have the number 16 and on the right side we have a number (which is unknown) under a square root,

Therefore we ask the question: "The square root of which number is 16?"

We can answer this question by guessing and checking the square roots of different numbers using a calculator, but a better way is to remember that square root and squaring are inverse operations and therefore:

$16=\sqrt{16^2}$

Therefore the answer to the above question is of course the number:

$16^2$

Let's calculate the numerical value of this term:

$16^2=256$

Therefore the answer to the above question, meaning - the unknown number under the square root in the problem, is the number 256:

$16 =\sqrt{\textcolor{red}{256}}$

Therefore the correct answer is answer B.