If 4$^{3x}$ = 16$^{x+1}$, find the value of x

Analyze the question: Both sides are powers of 4 (or of 2), so we can rewrite them with the same base and then equate the exponents.

Concept: If aᵐ = aⁿ, then m = n. Also, (aᵐ)ⁿ = aᵐⁿ.

Solution:

Write 16 as a power of 4: 16 = 4².

4^(3x) = 16^(x+1)

4^(3x) = (4²)^(x+1)

4^(3x) = 4^(2(x+1))

Since the bases are the same, equate the exponents:

3x = 2(x + 1)

3x = 2x + 2

x = 2

Check:

• Left side: 4^(3×2) = 4⁶ = 4096
• Right side: 16^(2+1) = 16³ = 4096 ✓

Both sides match, so x = 2, which is option A.

Tip: In index equations, first convert every number to the same base (often 2, 3, or 5). Then drop the bases and solve the simple equation formed by the exponents. Always substitute your answer back to check it.