Verify the Equality: 3^4 - (√25 - 2^2) - (2^3 - √4) Expression Analysis

To determine if the given equation is correct, we will simplify each expression in its sides separately,

This will be done while adhering to the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the term with the exponent - the leftmost term:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-(5-4)-(8-2)$We'll continue and finish simplifying the expressions inside the parentheses, meaning we'll perform the subtraction operations in them, then we'll perform the remaining subtraction operation:

$81-(5-4)-(8-2) =\\ 81-1-6=\\ 74$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-1-6=\\ 74$B. Let's continue with simplifying the expression on the right side of the given equation:

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4}$We'll start by simplifying the expression inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical values of the terms with exponents that are not in parentheses:

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} =\\ 81-5-(4-8)-2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the subtraction operation in it, then we'll perform the remaining subtraction operations:$81-5-(4-8)-2 =\\ 81-5-(-4)-2 =\\ 81-5+4-2 =\\ 78$Note that the result of the subtraction operation in the parentheses yielded a negative result and therefore in the next step we kept this result in parentheses and then applied the multiplication law stating that multiplying a negative number by a negative number gives a positive result (so ultimately we get an addition operation), then, we performed the addition and subtraction operations in the resulting expression,

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

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} =\\ 81-5-(-4)-2 =\\ 78$

Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})=3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} \\ \downarrow\\ 74=78$Obviously this equation does not hold, meaning - we got a false statement,

Therefore the correct answer is answer B.