Solving the Equation (x+2)^(2/3) = 9
This article will guide you through solving the equation (x+2)^(2/3) = 9. We'll break down the steps to find the solution for x.
Understanding the Equation
The equation involves a fractional exponent, which can be tricky to handle at first. Remember that a fractional exponent represents a combination of a root and a power. In this case, (2/3) means taking the cube root of the square of the expression (x+2).
Solving the Equation

Isolate the base: To start, we need to isolate the term with the exponent. Since the exponent is applied to the entire expression (x+2), we don't need to do anything extra here.

Raise both sides to the reciprocal power: The reciprocal of (2/3) is (3/2). Raising both sides of the equation to the power of (3/2) will eliminate the fractional exponent on the left side.
(x+2)^(2/3 * 3/2) = 9^(3/2)

Simplify: This simplifies to:
(x+2) = 9^(3/2)

Calculate the right side: 9^(3/2) can be calculated as the square root of 9 cubed, which is 27.
(x+2) = 27

Solve for x: Subtract 2 from both sides of the equation.
(x) = 25
Therefore, the solution to the equation (x+2)^(2/3) = 9 is x = 25.
Checking the Solution
We can always check our answer by plugging it back into the original equation.
(25 + 2)^(2/3) = 27^(2/3) = (27^(1/3))^2 = 3^2 = 9
This confirms that x = 25 is the correct solution.