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Solve for T: 2^T + 2^(T+1) = 24
Published on September 7, 2024
Algebra
Solve for T in the exponential equation 2^T + 2^(T+1) = 24. The solution reveals T to be 3.
Question
2^T + 2^(T+1) = 24 Then, T=?
Answer
Let’s break down this exponential equation step-by-step.
1. Factor out a common factor:
Notice that both terms on the left-hand side of the equation have a factor of 2<sup>T</sup>. Let’s factor that out:
2<sup>T</sup> (1 + 2<sup>1</sup>) = 24
2. Simplify:
Simplify the expression inside the parentheses:
2<sup>T</sup> (1 + 2) = 24
2<sup>T</sup> (3) = 24
3. Isolate the exponential term:
Divide both sides of the equation by 3:
2<sup>T</sup> = 8
4. Express the right-hand side with the same base:
Recognize that 8 can be expressed as 2<sup>3</sup>:
2<sup>T</sup> = 2<sup>3</sup>
5. Solve for T:
Since the bases are the same, we can equate the exponents:
T = 3
Therefore, T = 3