Given that t = $2 ^{-x}$, find $2 ^{x + 1}$ in terms of t.

Analyze the question: We are given t = 2^(-x) and must express 2^(x+1) using only t.

Concept: Two index laws are needed:

• a^(m+n) = a^m · a^n
• a^(-m) = 1/a^m

Solution:

1. Split the expression: 2^(x+1) = 2^x · 2^1 = 2 · 2^x

2. Rewrite the given statement using the negative index law: t = 2^(-x) = 1/2^x

3. Make 2^x the subject: 2^x = 1/t

4. Substitute back: 2^(x+1) = 2 · (1/t) = 2/t

Check: Let x = 1. Then t = 2^(-1) = 1/2, and 2^(x+1) = 2^2 = 4. Using option A: 2/t = 2/(1/2) = 4 ✓.

Tip: When you see a negative power such as 2^(-x), flip it to get the reciprocal, 2^x = 1/t. A quick substitution check with a simple value like x = 1 can confirm your answer in seconds during the exam.