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this question is pretty hard i dont understand how they got to that conclusion explain the steps etc

Let's break down the solution in the worked example step-by-step to clarify how they arrived at the final conclusion.

### Problem:
Solve for \( x \) in the equation:
\[
x^{\frac{4}{3}} = \frac{1}{81}
\]

### Step 1: Rewrite the equation
We start with the given equation:
\[
x^{\frac{4}{3}} = \frac{1}{81}
\]
This states that some power of \( x \), specifically \( x \) raised to the power \( \frac{4}{3} \), equals \( \frac{1}{81} \).

### Step 2: Eliminate the exponent on \( x \)
To solve for \( x \), we need to get rid of the exponent \( \frac{4}{3} \). We can do this by raising both sides of the equation to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \). This works because \( \left(x^{\frac{4}{3}}\right)^{\frac{3}{4}} = x \), as the exponents multiply and cancel out.

Apply this to both sides:
\[
\left(x^{\frac{4}{3}}\right)^{\frac{3}{4}} = \left(\frac{1}{81}\right)^{\frac{3}{4}}
\]
On the left side, the exponents cancel out:
\[
x = \left(\frac{1}{81}\right)^{\frac{3}{4}}
\]

### Step 3: Simplify the right-hand side
Now, we need to simplify \( \left(\frac{1}{81}\right)^{\frac{3}{4}} \).

First, recognize that \( 81 \) is a power of 3:
\[
81 = 3^4
\]
Thus, we can rewrite \( \frac{1}{81} \) as:
\[
\frac{1}{81} = 3^{-4}
\]

Substitute this back into the equation:
\[
x = \left(3^{-4}\right)^{\frac{3}{4}}
\]

Now, simplify the exponents by multiplying \( -4 \) and \( \frac{3}{4} \):
\[
x = 3^{-3} = \frac{1}{27}
\]

### Final Answer:
\[
x = \frac{1}{27}
\]

### Explanation Summary:
- The equation \( x^{\frac{4}{3}} = \frac{1}{81} \) is solved by raising both sides to the power of \( \frac{3}{4} \), which cancels the exponent on \( x \).
- Simplifying \( \left(\frac{1}{81}\right)^{\frac{3}{4}} \) involves recognizing that \( 81 = 3^4 \), which eventually leads to the solution \( x = \frac{1}{27} \).

Let me know if you'd like more clarification, or if you have any questions!

### Follow-up questions:
1. What other methods can be used to solve equations with fractional exponents?
2. How can we solve \( x^a = b \) if \( a \) is a decimal rather than a fraction?
3. What is the general rule for solving equations of the form \( x^n = c \)?
4. How would the solution change if the exponent were negative in the original equation?
5. What is the significance of simplifying exponents when solving these types of equations?

### Tip:
When dealing with exponents, converting numbers to powers of simpler numbers (like writing 81 as \( 3^4 \)) can make the problem much easier to manage.

Discover the method to solve the equation x^(4/3) = 1/81 with a detailed, step-by-step explanation. This tutorial is ideal for high school students learning about exponents and roots.

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