Solve the following equation with a rational exponent: 5x43=80

Question

5
x
4
3
=
80
.(1 point)
Responses

8
8

12
12

16
16

40.32

Answer 1

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

1. Isolate the term with \( x \): First, divide both sides by 5:

   \[ x^{4/3} = \frac{80}{5} = 16 \]

2. Remove the exponent: To isolate \( x \), raise both sides to the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \):

   \[ x = 16^{3/4} \]

3. Calculate \( 16^{3/4} \): We can break this down:

   \[ 16^{3/4} = (16^{1/4})^3 \]

   The fourth root of 16 is:

   \[ 16^{1/4} = 2 \]

   Now raise it to the third power:

   \[ (2)^3 = 8 \]

Thus, the solution to the equation \( 5x^{4/3} = 80 \) is:

\[ \boxed{8} \]