solve the following equation with a rational exponent: 2x{3/2}+=16

Question

A.no solution B.8 C.3 D.84

Answer 1

To solve the equation \( 2x^{3/2} = 16 \), follow these steps:

1. Divide both sides by 2:

   \[ x^{3/2} = \frac{16}{2} = 8 \]

2. Now, to eliminate the rational exponent, raise both sides to the power of \( \frac{2}{3} \):

   \[ x = 8^{2/3} \]

3. Calculate \( 8^{2/3} \). First, we can find the cube root of 8, which is 2:

   \[ 8^{1/3} = 2 \]

4. Then square it:

   \[ (8^{1/3})^2 = 2^2 = 4 \]

Thus, we find:

\[ x = 4 \]

Now let's check the options provided (A: no solution, B: 8, C: 3, D: 84). Since \( x = 4 \) is not listed among the options, there seems to be a discrepancy.

To summarize, the solution to the equation \( 2x^{3/2} = 16 \) is \( x = 4 \), but since this value is not in the given choices, the answer would lean towards option A: no solution from the given choices.

Answer 2

solve the following equation with a rational exponent: 5x{4/3}=80

Answer 3

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. Raise both sides to the power of \( \frac{3}{4} \) to eliminate the rational exponent: \[ x = 16^{3/4} \]

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

   • First, find the fourth root of 16: \[ 16^{1/4} = 2 \]
   • Then, raise that result to the power of 3: \[ (16^{1/4})^3 = 2^3 = 8 \]

Thus, the solution is: \[ x = 8 \]

So, the value of \( x \) is \( 8 \).