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2013 CEMC Gauss (Grade 8) Problems/Problem 12

Revision as of 14:27, 6 May 2025 by Anabel.disher (talk | contribs) (Created page with "== Problem == The value of <math>(2^3)^2 - 4^3</math> is <math> \textbf{(A)}\ 0 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12...")
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Problem

The value of $(2^3)^2 - 4^3$ is

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Solution 1

$(2^3)^2 - 4^3 = 8^2 - 4^3 = 64 - 64 = \boxed {\textbf {(A) } 0}$

~anabel.disher

Solution 2

We can notice that $(a^x)^y = a^{xy} = (a^y)^x$, and use the fact that $a - a = 0$ to get the answer:

$(2^3)^2 - 4^3 = (2^2)^3 - 4^3 = 4^3 - 4^3 = \boxed {\textbf {(A) } 0}$

~anabel.disher

Solution 3

$(2^3)^2 - 4^3 = 2^{3 \times 2} - 4^3 = 2^6 - 4^3 = 64 - 64 = \boxed {\textbf {(A) } 0}$

~anabel.disher

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