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

Question

Responses

16
16

12
12

8
8

40.32

Answer 1

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

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

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

3. Calculate \( 16^{3/4} \):

   • Break it down: \( 16^{3/4} = (16^{1/4})^3 \)
   • The fourth root of 16 is 2 (since \( 2^4 = 16 \)): \[ 16^{1/4} = 2 \]
   • Now raise that result to the power of 3: \[ 2^3 = 8 \]

Therefore, the solution to the equation is: \[ x = 8 \]

The correct response is 8.