Verify the Complex Expression: 2³·(6²-7·3)÷(√36-√1)+√4 = (4³-5²)÷(2+1)·2

In order to determine if the given equation is correct, we will simplify each of the expressions in 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:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4}$We'll start by simplifying the expressions inside the parentheses, this will be done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent), simultaneously we'll calculate the numerical value of the root in the second term from the left and the numerical value of the term with the exponent, which is the first multiplier from the left in the leftmost expression:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4} =\\ 8\cdot(36-21):(6-1)+2$Let's continue and perform the subtraction operations in the parentheses:

$8\cdot(36-21):(6-1)+2 =\\ 8\cdot15:5+2$Note that there is no defined order of operations between multiplication and division, and there are no parentheses in this expression defining precedence for either operation, therefore we'll calculate the result of the leftmost term (with all its operations) as we compute step by step from left to right, then we'll perform the addition operation:

$8\cdot15:5+2 =\\ 120:5+2 =\\ 24+2=\\26$We have completed simplifying the expression on the left side of the given equation, let's summarize the simplification steps:
$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4} =\\ 8\cdot(36-21):(6-1)+2 =\\ 8\cdot15:5+2 =\\ 24+2=\\26$

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

$(4^3-5^2):(2+1)\cdot2$

Similar to the previous part, we'll start by simplifying the expressions in parentheses, this will be done by calculating the numerical values of the terms with exponents, then we'll perform the addition and subtraction operations in the parentheses:

$$$(4^3-5^2):(2+1)\cdot2 =\\ (64-25):(2+1)\cdot2=\\ 39:3\cdot2$Note (again) that there is no defined order of operations between multiplication and division, and there are no parentheses in this expression defining precedence for either operation, therefore we'll calculate the result of the expression we got (with all its operations) as we compute step by step from left to right:

$39:3\cdot2 =\\ 13\cdot2=\\ 26$

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

$(4^3-5^2):(2+1)\cdot2 =\\ 39:3\cdot2 =\\ 26$ Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4}=(4^3-5^2):(2+1)\cdot2 \\ \downarrow\\ 26=26$We found that the equation is indeed true, meaning - we got a true statement,

Therefore the correct answer is answer A.