Verify the Equality: 4³ - (√49 + √64)·2² vs (4³ - √49) + √64·2²

In order to determine if the given equation is correct, we will simplify each of the expressions on its sides separately,

This will be done while following 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:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2$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, which states that a root is actually an exponent), simultaneously we'll calculate the numerical value of the term with the exponent, which is the multiplier to the right of the parentheses in the second expression from the left and the numerical value of the term with the exponent - the first from the left:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2 =\\ 64-(7+8)\cdot4=\\$We'll continue to perform the addition operation inside the parentheses, in the next step we'll calculate the multiplication by the second term from the left and finally we'll perform the subtraction operation:

$64-(7+8)\cdot4=\\ 64-15\cdot4=\\ 64-60=\\ 4$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2 =\\ 64-(7+8)\cdot4=\\ 64-60=\\ 4$B. Let's continue with simplifying the expression on the right side of the given equation:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2$Similar to the previous part, we'll start by simplifying the expression in parentheses, this is done by calculating the numerical values of the terms with exponents (and of course this includes the square root), then we'll perform the subtraction operation in the parentheses, simultaneously we'll calculate the numerical values of the root in the second term from the left and of the term with the exponent multiplying it:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 =\\ (64-7)+8\cdot4 =\\ 57 +8\cdot4$We'll continue and perform the multiplication in the second term from the left in the next step we'll perform the addition operation:

$57 +8\cdot4 =\\ 57+32=\\ 89$We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 =\\ (64-7)+8\cdot4 =\\ 57+32=\\ 89$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2=(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 \\ \downarrow\\ 4=89$Obviously this equation does not hold true, meaning - we got a false statement,

Therefore the correct answer is answer B.