(x+3)^5/(x+3)^2=64/27

2 min read Jun 16, 2024

Solving the Equation: (x+3)^5 / (x+3)^2 = 64/27

This problem involves simplifying expressions with exponents and solving for the unknown variable 'x'. Let's break it down step-by-step:

1. Simplifying the Expression

The left side of the equation has a common base of (x+3). Using the rule of exponents that states a^m / a^n = a^(m-n), we can simplify:

(x+3)^5 / (x+3)^2 = (x+3)^(5-2) = (x+3)^3

2. Rewriting the Equation

Now our equation looks like this:

(x+3)^3 = 64/27

3. Finding the Cube Root

The equation now shows a cube of (x+3) equaling 64/27. To isolate (x+3), we need to take the cube root of both sides:

∛[(x+3)^3] = ∛[64/27]

This simplifies to:

x+3 = 4/3

4. Solving for x

Finally, we isolate 'x' by subtracting 3 from both sides:

x = 4/3 - 3

5. The Solution

Calculating the result, we get:

x = -5/3

Therefore, the solution to the equation (x+3)^5 / (x+3)^2 = 64/27 is x = -5/3.