We are given the equation:

\[
\left(\frac{x}{3}\right)^4 = \frac{16}{81}
\]

To solve for \( x \), follow these steps:

### Step 1: Take the fourth root of both sides
Taking the fourth root will eliminate the exponent on the left-hand side.

\[
\frac{x}{3} = \pm \sqrt[4]{\frac{16}{81}}
\]

The fourth root of \( \frac{16}{81} \) can be computed as:

\[
\sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3}
\]

Thus, we have:

\[
\frac{x}{3} = \pm \frac{2}{3}
\]

### Step 2: Solve for \( x \)
Multiply both sides of the equation by 3 to isolate \( x \):

\[
x = \pm 2
\]

### Final Answer:
\[
x = 2 \quad \text{or} \quad x = -2
\]

Would you like further clarification on any of the steps?

Here are 5 related questions:
1. How do you solve similar equations with other exponents?
2. How do you solve equations involving square roots?
3. What are some common techniques for solving equations with rational exponents?
4. How do you apply the fourth root in more complex equations?
5. How can you verify your solutions in this type of equation?

**Tip:** Always remember to consider both the positive and negative roots when solving equations with even exponents.

Learn how to solve the exponential equation (x/3)^4 = 16/81 by finding the fourth root of both sides. This step-by-step guide covers key algebraic concepts, including exponents, roots, and simplifying fractions. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation