Solve the following equation with a rational exponent: 5x4/3=80.(1 point)

Question

Responses

16
16

8
8

12
12

40.32

Answer 1

To solve the equation \( 5x^{4/3} = 80 \), we can follow these steps:

1. Divide both sides by 5: \[ x^{4/3} = \frac{80}{5} \] \[ x^{4/3} = 16 \]

2. Now, to eliminate the rational exponent \( \frac{4}{3} \), raise both sides to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \): \[ x = 16^{\frac{3}{4}} \]

3. Calculate \( 16^{\frac{3}{4}} \): First, find \( 16^{\frac{1}{4}} \): \[ 16^{\frac{1}{4}} = 2 \quad (\text{since } 16 = 2^4) \] Then, raise 2 to the 3rd power: \[ 16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8 \]

4. Thus, we find: \[ x = 8 \]

So the solution to the equation \( 5x^{4/3} = 80 \) is \( \boxed{8} \).