To solve the equation \( 5x^{4/3} = 80 \), we will first isolate \( x^{4/3} \).
- Divide both sides by 5:
\[ x^{4/3} = \frac{80}{5} \]
\[ x^{4/3} = 16 \]
- Now we can eliminate the rational exponent by raising both sides to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \):
\[ x = 16^{\frac{3}{4}} \]
- Next, we need to compute \( 16^{\frac{3}{4}} \). First, we can break it down:
\[ 16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 \]
- Since \( 16 = 2^4 \), we have:
\[ 16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \]
- Now, substituting this back in, we get:
\[ x = (2)^3 = 8 \]
Thus, the solution to the equation \( 5x^{4/3} = 80 \) is:
\[ \boxed{8} \]
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