Compare Complex Expressions: (9²-2√81)+4³÷2³ vs (8²+2√64)-3²

Simplify each of the following expressions separately :

a. Let's start with the expression on the left:

$(9^2-2\sqrt{81})+4^3:2^3$

Note that in this expression the parentheses are meaningless given that no mathematical operation is applied to them, therefore we can remove them, whilst writing the division operation as a fraction:

$(9^2-2\sqrt{81})+4^3:2^3=\\ 9^2-2\sqrt{81}+\frac{4^3}{2^3}$

Note that the number 4 is a power of the number 2:

$4=2^2$

Therefore, we can have all terms with identical bases in the fraction and use the power rule for dividing terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Remember the power rule for power of a power:

$(a^m)^n=a^{m\cdot n}$

Apply these two rules in handling the fraction in the expression mentioned above (which is the third term from the left):

$\frac{4^3}{2^3} =\frac{(2^2)^3}{2^3} =\frac{2^{2\cdot3}}{2^3} =\frac{2^6}{2^3}=2^{6-3} =2^3=8$

Where in the first stage we replaced the number 4 with a power of 2, in the next stage we applied the power rule for power of a power mentioned earlier, and then we simplified the expression in the fraction's numerator and applied the power rule for dividing terms with identical bases. We completed the calculation by simplifying the resulting expression,

Let's return to the expression and complete its calculation by substituting what we've obtained so far:

$9^2-2\sqrt{81}+\frac{4^3}{2^3} \\ \downarrow\\ 9^2-2\sqrt{81}+8=\\ 81-2\cdot9+8=\\ 81-18+8=71$

b. Let's continue with the expression on the right:

$(8^2+2\sqrt{64})-3^2$

Here too we notice that the parentheses are meaningless since no mathematical operation is applied to them, therefore, similar to the previous part, we'll remove them:

$8^2+2\sqrt{64}-3^2$

notice that in this expression there's no special use of power rules therefore we'll continue with direct simplification through numerical calculation only:

$8^2+2\sqrt{64}-3^2 =\\ 64+2\cdot8-9=\\ 64+16-9=71$

Let's return to the original problem and substitute the results of both solution parts detailed so far in a and b in place of the expressions on the right and left:

$(9^2-2\sqrt{81})+4^3:2^3\text{ }\textcolor{red}{_{——\text{ }}}(8^2+2\sqrt{64})-3^2 \\ \downarrow\\ 71\text{ }\textcolor{red}{_{——\text{ }}}71$

Therefore it's clear that equality exists, meaning that:

$71\text{ }\textcolor{red}{{=\text{ }}}71$

The correct answer is answer c.