SAT Math Problem (8)

Question

If n is a positive integer and m = 2ⁿ⁺² + 2ⁿ , what is 2ⁿ⁺³ in terms of m?

Answer 1

You need to manipulate the equation to isolate 2ⁿ⁺³ on one side of the equals sign.

 

m = 2ⁿ⁺² + 2ⁿ

m = 2ⁿ•2² + 2^{n   ... reverse of product of powers property (laws of exponents)}  

m = 2ⁿ•4 + 2ⁿ

m = 2ⁿ(4+1)    ... factor out 2ⁿ

m = 2ⁿ•5  

m/5 = 5•2ⁿ/5   .... divide both sides by 5

2ⁿ = m/5    ... symmetric property

2^3•2ⁿ = 2³•m/5  .., multiply both sides by 2³

2ⁿ⁺³ = 8m/5   ... product of powers

Answer 2

Given: m = 2ⁿ⁺² + 2ⁿ; this means m = (2ⁿ)(2²+ 1) = 5(2ⁿ).

 

Therefore m = 5(2ⁿ),

m/5 = 2ⁿ and if you multiply both sides of this by 2² you get    (4/5)m = 2ⁿ⁺³