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

Calculate the values of all terms in the equation using direct computation:

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

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

Note that this is an equation and therefore we can move terms from one side to the other, doing so whilst 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 that we obtained:

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:

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

The correct answer is answer B.