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The equation to solve is \( 5x^{\frac{4}{3}} = 80 \).

Let's solve it step by step:

### Step 1: Isolate \( x^{\frac{4}{3}} \)
Divide both sides of the equation by 5:
\[
x^{\frac{4}{3}} = \frac{80}{5} = 16
\]

### Step 2: Eliminate the exponent
Raise both sides to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \):
\[
x = 16^{\frac{3}{4}}
\]

### Step 3: Calculate \( 16^{\frac{3}{4}} \)
This expression can be broken down as:
\[
16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3
\]
Since \( 16^{\frac{1}{4}} = 2 \), we have:
\[
x = 2^3 = 8
\]

So, the solution is \( x = 8 \).

Do you have any questions, or would you like further explanation? Here are five related questions to explore:

1. What is the general method for solving equations with fractional exponents?
2. How do you simplify expressions like \( 16^{\frac{3}{4}} \)?
3. How can you verify solutions for equations with exponents?
4. What other methods can be used to solve \( x^{\frac{m}{n}} = a \)?
5. How would the approach change if the exponent were negative?

**Tip:** Always check your solution by substituting it back into the original equation.

Solve the following equation with a rational exponent: 5x^(4/3) = 80.

Step-by-step solution for solving the equation 5x^(4/3) = 80. This algebra problem involves rational exponents and simplification techniques, suitable for high school students in Grades 9-12.

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