solve the following equation with a rational exponent 2x ^ 2/3 = 16

Answer 1

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

1. Isolate \( x^{2/3} \): \[ x^{2/3} = \frac{16}{2} \] \[ x^{2/3} = 8 \]

2. Remove the rational exponent: Raise both sides to the power of \( \frac{3}{2} \) to eliminate the exponent on \( x \): \[ (x^{2/3})^{3/2} = 8^{3/2} \] \[ x^1 = 8^{3/2} \] \[ x = 8^{3/2} \]

3. Calculate \( 8^{3/2} \): First, compute \( 8^{1/2} \): \[ 8^{1/2} = \sqrt{8} = 2\sqrt{2} \]

   Now raise it to the 3rd power: \[ (2\sqrt{2})^3 = 2^3 \cdot (\sqrt{2})^3 = 8 \cdot 2\sqrt{2} = 16\sqrt{2} \]

So, the solution to the equation \( 2x^{2/3} = 16 \) is: \[ x = 16\sqrt{2} \]