https://file.aiquickdraw.com/mathbot/file/1727456021369r0vdm3gq.jpg

The given equation is:

\[
x^{\frac{4}{3}} = \frac{1}{256}
\]

### Step-by-step Solution:

1. To solve for \( x \), first isolate the variable by raising both sides of the equation to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \). This will cancel out the exponent on the left-hand side:

\[
\left( x^{\frac{4}{3}} \right)^{\frac{3}{4}} = \left( \frac{1}{256} \right)^{\frac{3}{4}}
\]

This simplifies to:

\[
x = \left( \frac{1}{256} \right)^{\frac{3}{4}}
\]

2. Now, simplify \( \left( \frac{1}{256} \right)^{\frac{3}{4}} \). First, rewrite \( 256 \) as \( 2^8 \), since \( 256 = 2^8 \). The equation becomes:

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

This simplifies to:

\[
x = \frac{1}{2^6} = \frac{1}{64}
\]

### Final Answer:

\[
x = \frac{1}{64}
\]

Would you like any further details or clarification?

Here are some related questions:
1. What happens if the exponent was a negative fraction instead?
2. How would you solve for \( x \) if the right-hand side were a fraction other than \( \frac{1}{256} \)?
3. What if the base on the left-hand side was not \( x \), but a different term like \( 2x \)?
4. Can you express \( 256 \) in other base forms to solve similar problems?
5. What if the equation was \( x^{\frac{4}{3}} = 256 \) instead?

**Tip:** Always try simplifying the powers of numbers as much as possible when dealing with exponents to make solving easier.

Learn how to solve the exponent equation x^(4/3) = 1/256 with a detailed step-by-step solution. This problem covers algebraic concepts, exponent rules, and the simplification of powers of numbers. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation