Solve: Square Root of 64/10000 Plus 92/100 Equals x^450

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 an exponent applied to terms in parentheses:

$\big(\frac{a}{c}\big)^n=\frac{a^n}{c^n}$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 will therefore use this understanding and simplify the expression in the left side, by applying the law of exponents mentioned in b to the first term:

$\sqrt{\frac{64}{10000}}+\frac{92}{10^2}=\textcolor{red}{☐}^{450} \\ \frac{\sqrt{64}}{\sqrt{10000}}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\ \frac{8}{100}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\$In the first stage, we applied the square root (which is actually a half power) to both numerator and denominator of the fraction in the first term on the left, this is according to the law of exponents mentioned in b and in the next stage we calculated the numerical value of the roots in the numerator and denominator of the fraction,

We'll continue and simplify the second term from the left on the left side, meaning we'll calculate the numerical value of the expression in the fraction's denominator, then we'll perform the addition operation between the two resulting fractions:

$\frac{8}{100}+\frac{92}{10^2} =\textcolor{red}{☐}^{450} \\ \frac{8}{100}+\frac{92}{100} =\textcolor{red}{☐}^{450} \\ \frac{8+92}{100} =\textcolor{red}{☐}^{450} \\ \frac{100}{100} =\textcolor{red}{☐}^{450} \\ 1 =\textcolor{red}{☐}^{450}$In the second stage we performed the addition operation between the two fractions, using the fact that both denominators are identical (and therefore there was no need to expand them, but it was sufficient to combine them into one fraction using the identical denominator, which is actually the common denominator) then we simplified the expression in the numerator of the resulting fraction on the left side and finally remembered that dividing any number by itself will always give the result 1,

Now let's examine the equation we got in the last stage and answer the question that was asked,

On the left side we got the number 1, and on the right side there is a number (unknown) raised to the power of 450,

Remember that raising the number 1 to any power will always give the number 1, therefore the answer is the number 1,

Meaning:

$1 =\textcolor{red}{1}^{450}$However, since this is an even power (power of 450), we must also consider the negative possibility,

Meaning it also holds that:

$1=\textcolor{red}{(-1)}^{450}$

Therefore the correct answer is answer a.