Evaluate 5³÷(4²+3²)-(√100-8²): Order of Operations Challenge

To determine if the given equality is correct we will simplify each of the expressions that appear in it separately,

This is done while keeping in mind the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all,

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

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)$We start by simplifying the expressions inside the parentheses, this is done by calculating their numerical value (while remembering the definition of the square root as the non-negative number whose square gives the number under the root), in parallel we calculate the numerical value of the other terms in the expressions:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64)$We continue and finish simplifying the expressions inside the parentheses, meaning we perform the subtraction operation in them, then we perform the division operation which is in the first term from the left and then the remaining subtraction operation:

$125:(16+9)-(10-64) =\\ 125:25-(-54) =\\ 5+54 = 59$We note that the result of the subtraction operation in the parentheses is a negative result and therefore in the next step we will leave this result in the parentheses and then apply the multiplication law which states that multiplying a negative number by a negative number will give a positive result (so that in the end an addition operation is obtained), then, we perform the addition operation in the expression that was obtained,

We finished simplifying the expression on the left side of the given equality, let's summarize the simplification steps:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64) =\\ 5+54 =\\ 59$

B. We continue from simplifying the expression on the right side of the given equality:

$5^3:4^2+3^2-\sqrt{100}+8^2$We recall again the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all, and note that although in this expression there are no parentheses, there are terms in fractions and a division operation, so we start by calculating their numerical value, then we perform the division operation:

$5^3:4^2+3^2-\sqrt{100}+8^2 =\\ 125:16+9-10+64 =\\ 7\frac{13}{16}+9-10+64=\\ 70\frac{13}{16}$We note that since the division operation that was performed in the first term from the left yielded an incomplete result (greater than the divisor), we marked this result as a mixed number, then we performed the remaining addition and subtraction operations,

We finished simplifying the expression on the right side of the given equality, the simplification of this expression is short, so there is no need to summarize,

Let's go back now to the given equality and place in it the results of simplifying the expressions that were detailed in A and B:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)=5^3:4^2+3^2-\sqrt{100}+8^2 \\ \downarrow\\ 59= 70\frac{13}{16}$As can be seen this equality does not hold, meaning - we got a false sentence,

So the correct answer is answer B.