An Exponent Tower

Algebra Level 1

True or False?

For positive reals $a, b, c,$

$\large \left( a^ {\color{#D61F06}b}\right) ^ {\color{#3D99F6}c} = \left( a ^ {\color{#3D99F6}c} \right)^ {\color{#D61F06}b}.$

True False

6 solutions

Yee-Lynn Lee
Sep 6, 2015

$(a^{b})^{c}=a^{b \times c}$

$(a^{c})^{b}=a^{c \times b}$

By the commutative property of multiplication, $b \times c=c \times b$ Therefore, $\boxed{(a^{b})^{c}=(a^{c})^{b}}$

Sidenote: Be careful with the parenthesis (brackets). The expression $\large{a^{ b^{ c } }}$ is not necessarily equal to $\large{a^{ c^{ b }}}$ .

Arulx Z - 5 years, 9 months ago

Assuming a=1 b=2 and c=3

(1^2)^3 = 8 (1^3)^2 = 9

This is not always true. Brackets always come first. Incorrect answer

Zach Ross-Clyne - 5 years, 9 months ago

dude 1 ^ 2 = 1 . I think you wanted to say 2 ^2 ^3

atharv toraskar - 5 years, 8 months ago

Zach, it is true that brackets come first. Even with your hypothetical values for a, b, and c, this equation holds true.

For the left side, doing brackets first: 1^2 = 1. Then, doing exponents: 1^3 = 1. For the right side, doing brackets first: 1^3 = 1. Then, doing exponents. 1^2 = 1.

1 = 1. This is always true. Correct answer.

Alec Camhi - 5 years, 9 months ago

Shivangi Gupta
Oct 20, 2015

(a^b)^c=a^(b c) (a^c)^b=a^(c b) on comparing powers b c=c b

Rajat Kasera
Sep 13, 2015

(a^b)^c =a^b×c (a^c)^b= a^c×b

Hadia Qadir
Sep 13, 2015

True Both equals (a)^bc

Sadasiva Panicker
Sep 13, 2015

(a^b)^c = a^bc & (a^c)^b =a^cb Therefore (a^b)c =(a^c)b

Leonardo Fajardo
Sep 6, 2015

communtative property bxc=cxb

An Exponent Tower

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