Exponent Simplification

Algebra Level 3

$\Large{\text{Is }\left(2^\color{#D61F06}{a}\right)^{\color{#3D99F6}{b}} \text{ } = \text{ } 2^{\color{#D61F06}{a}+\color{#3D99F6}{b}} \quad \text{ or } \quad 2^{\color{#D61F06}{a}\color{#3D99F6}{b}} \text{ ?}}$

To see which simplification is correct, you plug in specific numbers $\color{#D61F06}{a}=A$ and $\color{#3D99F6}{b}=B$ , but somehow both proposed simplifications give the same answer: $64.$ Find $\left(A-B\right)^2.$

The answer is 12.

2 solutions

Eli Ross Staff
Sep 14, 2015

Since $2^6 = 64,$ we must have $A+B = AB = 6,$ so $(A-B)^2 = (A+B)^2 - 4AB = 6^2 - 4\cdot 6 = 12.$

Identical approach :)

Mehul Arora - 5 years, 9 months ago

Did the exact same way.

Aniruddha Bagchi - 4 years, 11 months ago

Oli Hohman
Sep 16, 2015

2^6 = 64, so you have A+B=AB=6 Using substitution, you're left with the quadratic B^2-6B+6=0 Solve this with the quadratic formula to find both values of A and B. You get two values 3+sqrt(3) and 3-sqrt(3). Plugging these into (A-B)^2 yields (2sqrt(3))^2. = 12

Exponent Simplification

The answer is 12.

2 solutions

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