3^t * 9^(t + 3) = 27^2

Question

How do I find the value of t?

Answer 1

The equation is

Rewrite 9 as 3²  and 27 as 3³ .

Power of a power property

Product of powers property

Since the bases are the same, equate the powers and solve.

Solution t = 0.

Answer 2

The exponential equation is 3^t * 9^{(t + 3)} = 27² .

3^t * (3²)^{(t + 3)} = (3³)²

Power of power property : (a^b)^c = a^bc, where a, b, and c are all real numbers.

3^t * 3^{2(t + 3)} = 3^{3 * 2}

3^t * 3^{2t + 6} = 3⁶

Product of powers property : a^b × a^c = a^{(b + c)} , where a, b, and c are all real numbers.

3^{t + 2t + 6} = 3⁶

3^{3t + 6} = 3⁶

Property of equality for exponential functions : If a^x = a^y , then x = y, where a, x, and y are all real numbers.

3t + 6 = 6

3t = 6 - 6 = 0

⇒ t = 0.

The value of t = 0.