(1/16)^3x=64^2(x+8)

2 min read Jun 16, 2024

Solving the Exponential Equation: (1/16)^3x = 64^2(x+8)

This article will guide you through solving the exponential equation (1/16)^3x = 64^2(x+8). We will utilize the properties of exponents to simplify the equation and solve for x.

Step 1: Expressing Bases as Powers of the Same Number

The first step is to express both 1/16 and 64 as powers of the same base.

1/16 can be written as 4^-2 (since 1/16 = 1/4^2 = 4^-2).
64 can be written as 4^3 (since 64 = 4^3).

Substituting these values into the original equation, we get:

(4^-2)^3x = (4^3)^2(x+8)

Step 2: Simplifying using Exponent Rules

Using the rule (a^m)^n = a^(m*n), we can simplify the equation further:

4^(-6x) = 4^(6(x+8))

Step 3: Solving for x

Now that both sides of the equation have the same base, we can equate the exponents:

-6x = 6(x+8)

Expanding the right side of the equation:

-6x = 6x + 48

Combining like terms:

-12x = 48

Dividing both sides by -12:

x = -4

Solution

Therefore, the solution to the exponential equation (1/16)^3x = 64^2(x+8) is x = -4.