Question

Solve the following equation with a rational exponet: 2x3/2=16.

Answer 1

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

1. **Isolate the term with the exponent**:
\[
x^{3/2} = \frac{16}{2}
\]
\[
x^{3/2} = 8
\]

2. **Raise both sides to the power of \( \frac{2}{3} \)** to eliminate the exponent:
\[
\left( x^{3/2} \right)^{\frac{2}{3}} = 8^{\frac{2}{3}}
\]

This simplifies to:
\[
x = 8^{\frac{2}{3}}
\]

3. **Calculate \( 8^{\frac{2}{3}} \)**:
- First, find the cube root of 8: \( 8^{1/3} = 2 \).
- Then, square the result: \( 2^2 = 4 \).

Therefore,
\[
x = 4
\]

4. **Check the solution**: Substitute \( x = 4 \) back into the original equation:
\[
2(4)^{3/2} = 16
\]
Calculate \( (4^{3/2}) \):
\[
4^{3/2} = (4^{1/2})^3 = 2^3 = 8
\]
Then,
\[
2 \times 8 = 16,
\]
which matches the original equation.

Thus, the solution is
\[
\boxed{4}.
\]