Verify the Equality: 4³ + (√49 + √64) + 2² vs. Alternative Grouping

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 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:

$4^3+(\sqrt{49}+\sqrt{64})+2^2$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical value 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 terms with exponents, both the third term from the left and the first term from the left:

$4^3+(\sqrt{49}+\sqrt{64})+2^2 =\\ 64+(7+8)+4=\\$Note that the parentheses in this problem have no significance, as all operations between the different numerical terms are addition operations, however, parentheses determine operation precedence and therefore we'll first complete calculating the expression inside the parentheses and only then perform the addition operations (of course, in this case, this order of operations will yield the same result as if we removed the parentheses and calculated the sum of all terms):

$64+(7+8)+4=\\ 64+15+4=\\ 83$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$4^3+(\sqrt{49}+\sqrt{64})+2^2 =\\ 64+(7+8)+4=\\ 83$B. Let's continue with simplifying the expression on the right side of the given equation:

$(4^3+\sqrt{49})+\sqrt{64}+2^2$We'll begin again by simplifying the expressions inside the parentheses, this is done by calculating the numerical value 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 terms with exponents, both the third term from the left and the second term from the left:

$(4^3+\sqrt{49})+\sqrt{64}+2^2 =\\ (64+7)+8+4$Similar to the previous part, note that the parentheses in this problem have no significance, as all operations between the different numerical terms are addition operations, however, again we'll point out that parentheses determine operation precedence and therefore we'll first complete calculating the expression inside the parentheses and only then perform the addition operations:

$(64+7)+8+4 =\\ 71+8+4=\\ 83$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}+2^2 =\\ (64+7)+8+4 =\\ 83$$$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$4^3+(\sqrt{49}+\sqrt{64})+2^2=(4^3+\sqrt{49})+\sqrt{64}+2^2 \\ \downarrow\\ 83=83$We found that this equation indeed holds true, meaning - we got a true statement,

Therefore the correct answer is answer A.