Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2013 CEMC Gauss (Grade 8) Problems/Problem 12

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_CEMC_Gauss_(Grade_8)_Problems/Problem_12&oldid=248186"