Verify the Equality: √36 - (4² - 9) + √4 = √(25/10000) + 95/100

In order to determine the correctness (or incorrectness) of the given equation, let's simplify the expressions on both sides separately:

A. Let's start with the expression on the left side:
$\sqrt{36}-(4^2-9)+\sqrt{4}$Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and parentheses come before all, therefore we'll start by simplifying the expression in parentheses by calculating the numerical value of the term with the exponent inside them, then we'll perform the subtraction operation in the parentheses:

$\sqrt{36}-(4^2-9)+\sqrt{4} =\\ \sqrt{36}-(16-9)+\sqrt{4} =\\ \sqrt{36}-7+\sqrt{4}$Next, we'll calculate the numerical value of the roots in the expression (which are exponents in every way) and finally we'll perform the result of the expression combining addition and subtraction:

$\sqrt{36}-7+\sqrt{4}=\\ 6-7+2=\\ 1$We have completed simplifying the expression on the left side of the given equation, let's summarize the simplification process:

$\sqrt{36}-(4^2-9)+\sqrt{4} =\\ 6-7+2=\\ 1$

B. Let's continue with simplifying the expression on the right side of the given equation:

$\sqrt{\frac{25}{10000}}+\frac{95}{100}$For this, let's recall two laws of exponents:

B.1. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$B.2. The law of exponents for an exponent applied to parentheses containing a product of terms:

$\big(\frac{a}{c}\big)^n=\frac{a^n}{c^n}$Unlike in previous questions, we will not convert the square root to a power of one-half, but rather understand and internalize that according to the law of exponents mentioned in B.1. - a root is an exponent in every way and therefore all laws of exponents apply to it,

Let's return to the expression in question and apply this understanding to the first term on the left, which is the term with the root. Note that in this term, the power of one-half (meaning - the exponent equivalent to the square root) applies to the entire fraction under the root, therefore despite the absence of parentheses in the expression, we'll treat the fraction under the root as a fraction within parentheses with the power of one-half (of the root) applied to it, and therefore we'll apply the law of exponents mentioned in B.2. to this term, meaning - we'll apply the root to both the numerator and denominator of the fraction:

$\sqrt{\frac{25}{10000}}+\frac{95}{100} =\\ \frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100}$Let's continue and calculate the numerical value of the roots in the numerator and denominator of the fraction, then perform the addition operation between the fractions and simplify the resulting expression:

$\frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100} =\\ \frac{5}{100}+\frac{95}{100} =\\ \frac{5+95}{100}=\\ \frac{100}{100} =\\ 1$We performed the addition of fractions directly by putting them on one fraction line and adding the numerators (since the denominators in both fractions are identical, it is the common denominator, so there was no need to expand them), then we used the fact that dividing any number by itself always gives the result 1.

We have completed simplifying the expression on the right side of the given equation, let's summarize the simplification process:

$\sqrt{\frac{25}{10000}}+\frac{95}{100} =\\ \frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100} =\\ \frac{5}{100}+\frac{95}{100} =\\ 1$

Let's now return to the equation given in the problem and substitute the expressions on the left and right sides with the results of the simplifications detailed in A and B above, in order to determine the correctness (or incorrectness) of the given equation:

$\sqrt{36}-(4^2-9)+\sqrt{4}=\sqrt{\frac{25}{10000}}+\frac{95}{100} \\ \downarrow\\ 1=1$ We can now definitively determine that the given equation is indeed correct, meaning we have a true statement,

Therefore the correct answer is answer A.