Question

Solve the equation by expressing each side as a power of the same base and then equating exponents.

16^x+8 = 64^x-2

Answer 1

x = 22

Explanation:

1. Rewrite each base (16 and 64) as a power of the same base:

16=4², 64=4³ or 16=2⁴, 64=2⁶

`16^(x+8)=64^(x-2)= > (4^2)^(x+8)=(4^3)^(x-2)`

2. Multiply the powers:

2(x+8) = 2x+16 and 3(x-2) = 3x-6

`(4^2)^(x+8)=(4^3)^(x-2)= > 4^(2x+16)=4^(3x-6)`

3. Now that the bases are the same, set the exponents = each other and solve: 2x+16 = 3x-6

2x-2x+16+6 = 3x-2x-6+6

22 = x

Check: 16²²⁺⁸ = 16³⁰ = 1.329x10³⁶

64²²⁻² = 64²⁰ = 1.329x10³⁶