Solve the following equation with a rational exponent 2x3/2 = 16

Answer 1

To solve the equation $2x^{3/2}=16$, we will follow these steps:

1. Isolate the term with the exponent: $$x^{3/2}=\frac{16}{2}$$ Simplifying the right side gives: $$x^{3/2}=8$$

2. 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: $$x^{(3/2)\cdot (2/3)}=x^1=x$$

3. 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: $$\boxed{{\displaystyle 4}}$$