Solve for Missing Square Root: (2^5+18)÷(√5·√2)² Challenge

Let's recall two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for exponents within parentheses where terms are multiplied, but in the opposite direction:

$x^n\cdot y^n =(x\cdot y)^n$Unlike previous questions where we converted the square root to its corresponding half power according to the definition of root as an exponent, in this problem we will not make this conversion but rather understand and internalize that according to the definition of root as an exponent, the root is an exponent in every way and therefore all laws of exponents apply to it, particularly the law of exponents mentioned in b',

We'll start then by simplifying the expression in the right parentheses (as written there, actually this is the denominator of the fraction equivalent to the division operation, we'll emphasize this later), we'll simplify this expression while applying the above understanding and using the law of exponents mentioned in b':

$(\sqrt{5}\cdot\sqrt{2})^2=(\sqrt{5\cdot2})^2=(\sqrt{10})^2$where in the first stage we applied the above understanding and treated the square root (which is actually a half power) as an exponent in every way and applied the law of exponents mentioned in b', we applied this law in the direction specified there by noting that the two multiplication terms have the same exponent (half power - of the square root) and therefore we could combine them together as multiplication between the bases under the same root (same exponent), in the next stage we simplified the expression under the root,

Now we'll remember that square root and square power are inverse operations and therefore cancel each other out, therefore:

$(\sqrt{10})^2 =10$Now we'll return to the complete expression on the right side of the equation and insert this information, in the next stage we'll write the division operation as a fraction:

$\sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{5}\cdot\sqrt{2})^2 \\ \downarrow\\ \sqrt{\textcolor{red}{☐}}=(2^5+18):10\\ \sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10}$$$Note that the division operation acts on the parentheses and therefore refers to them in their entirety, therefore the entire expression in parentheses enters the numerator of the fraction,

Let's continue and simplify the expression in the numerator of the fraction on the right side:

$\sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{32+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{\not{50}}{\not{10}} \\ \sqrt{\textcolor{red}{☐}}=5$where in the first stage we calculated the numerical value of the term with the exponent in the numerator of the fraction and in the next stage we performed the addition operation in the numerator, in the final stage we reduced the resulting fraction (performing the division operation in fact),

Let's summarize the solution steps so far, we got that:

$\sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{5}\cdot\sqrt{2})^2 \\ \sqrt{\textcolor{red}{☐}}=(2^5+18):(\sqrt{10})^2 \\ \sqrt{\textcolor{red}{☐}}=(2^5+18):10\\ \downarrow\\ \sqrt{\textcolor{red}{☐}}=\frac{2^5+18}{10} \\ \sqrt{\textcolor{red}{☐}}=\frac{\not{50}}{\not{10}} \\ \sqrt{\textcolor{red}{☐}}=5$

Let's examine now the expression we got:

On the right side we got the number 5, and on the left side there's a (unknown) number that's under a square root,

In other words - we need to answer the question - "The square root of which number is 5?"

The answer to this question is of course the number: 25, it's easy to verify this by calculating its square root which indeed gives the result 5,

Meaning - for the equation to be true, under the root must be the number 25:

$\sqrt{\textcolor{red}{25}}=5$Therefore the correct answer is answer d.