Question

solve the equation with a rational exponent: x2/3 = 64

Answer 1

To solve the equation \( x^{2/3} = 64 \), follow these steps:

1. **Isolate \( x \)**:
Start by raising both sides of the equation to the power of \( \frac{3}{2} \) to eliminate the rational exponent.

\[
\left( x^{2/3} \right)^{3/2} = 64^{3/2}
\]

2. **Simplify the left side**:
The left side simplifies as follows:
\[
x^{(2/3) \cdot (3/2)} = x^1 = x
\]

3. **Calculate the right side**:
Now we need to calculate \( 64^{3/2} \):
- First, find \( \sqrt{64} \):
\[
\sqrt{64} = 8
\]
- Next, calculate \( 8^3 \):
\[
8^3 = 512
\]

4. **Set the equation**:
So, we have:
\[
x = 512
\]

5. **Conclusion**:
The solution to the equation \( x^{2/3} = 64 \) is:
\[
x = 512
\]