To solve the equation \( 2x^{3/2} = 16 \), we will follow these steps:
-
Isolate the term with the exponent: \[ x^{3/2} = \frac{16}{2} \] Simplifying the right side gives: \[ x^{3/2} = 8 \]
-
Eliminate the rational exponent: To eliminate the exponent \( \frac{3}{2} \), we can raise both sides of the equation to the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \): \[ \left( x^{3/2} \right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \] Simplifying the left side gives: \[ x^{(3/2) \cdot (2/3)} = x^1 = x \]
-
Calculate the right side: We need to calculate \( 8^{\frac{2}{3}} \): \[ 8^{\frac{2}{3}} = (8^{1/3})^2 \] Since \( 8^{1/3} = 2 \) (the cube root of 8 is 2), we square it: \[ (2)^2 = 4 \]
Putting it all together, we find that: \[ x = 4 \]
Thus, the solution to the equation \( 2x^{3/2} = 16 \) is: \[ \boxed{4} \]
5 answers