Question

Use like bases to solve the exponential equation. (164)3π⋅8=27 Enter the exact answer.

Answer 1

The exact solution to the exponential equation
`(1/64)^(3n).8 = 2^7` is n = -2/9.

How to solve the exponential equation

To solve the exponential equation
`(1/64)^(3n).8 = 2^7`, rewrite the bases using the same exponent.

First, simplify the equation. We can express 8 as
`2^3`:

`(1/64)^(3n).8 = 2^7\\(1/64)^(3n).2^3 = 2^7`

Now, rewrite (1/64) using the base 2:

`(2^-6)^(3n).2^3 = 2^7`

Using the property of exponents, simplify further:

`2^(-6 * 3n).2^3 = 2^7`

Now, equate the exponents on both sides:

-18n + 3 = 7

Solve for n:

-18n = 7 - 3

-18n = 4

n = 4 / -18

n = -2/9

Therefore, the exact solution to the exponential equation
`(1/64)^(3n).8 = 2^7` is n = -2/9.

Use like bases to solve the exponential equation. (1/64)^3n⋅8=2^7 Enter the exact answer.