To solve the equation \((x + 3)^{\frac{2}{3}} = 8\), we can follow these steps:

1. **Isolate the term with the exponent:**

   \[
   (x + 3)^{\frac{2}{3}} = 8
   \]

2. **Raise both sides to the power of \(\frac{3}{2}\) to eliminate the fractional exponent.** By doing this, the left-hand side will become a simple expression:

   \[
   \left[(x + 3)^{\frac{2}{3}}\right]^{\frac{3}{2}} = 8^{\frac{3}{2}}
   \]

   Simplifying the left-hand side:

   \[
   x + 3 = 8^{\frac{3}{2}}
   \]

3. **Calculate \(8^{\frac{3}{2}}\):**

   \[
   8^{\frac{3}{2}} = \left(8^1\right)^{\frac{3}{2}} = \left(2^3\right)^{\frac{3}{2}} = 2^{3 \times \frac{3}{2}} = 2^4 = 16
   \]

   So, we have:

   \[
   x + 3 = 16
   \]

4. **Solve for \(x\):**

   \[
   x = 16 - 3
   \]

   \[
   x = 13
   \]

Thus, the solution to the equation \((x + 3)^{\frac{2}{3}} = 8\) is \(\boxed{x = 13}\).

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve equations with other fractional exponents?
2. What are the properties of fractional exponents?
3. Can this method be used for negative numbers inside the parentheses?
4. What is the geometric interpretation of fractional exponents?
5. How do you check if a solution satisfies the original equation?

**Tip:** When working with fractional exponents, remember that the numerator indicates the power, and the denominator indicates the root.

Learn how to solve the equation (x + 3)^(2/3) = 8 with a detailed step-by-step solution. This problem involves exponents and equations, covering exponentiation and the power rule. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation