To solve the equation [tex]\(2 x^{\frac{3}{2}} = 16\)[/tex], we'll follow these steps:
1. Isolate the term involving the variable:
[tex]\[
2 x^{\frac{3}{2}} = 16
\][/tex]
2. Divide both sides by 2 to simplify the equation:
[tex]\[
x^{\frac{3}{2}} = \frac{16}{2}
\][/tex]
[tex]\[
x^{\frac{3}{2}} = 8
\][/tex]
3. Undo the exponent [tex]\(\frac{3}{2}\)[/tex] by raising both sides to the power of [tex]\(\frac{2}{3}\)[/tex], since [tex]\(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = x^{1} = x\)[/tex]:
[tex]\[
\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}
\][/tex]
[tex]\[
x = 8^{\frac{2}{3}}
\][/tex]
4. Simplify [tex]\(8^{\frac{2}{3}}\)[/tex]:
[tex]\[
8^{\frac{2}{3}} = \left(2^3\right)^{\frac{2}{3}}
\][/tex]
Using the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(2^3\right)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4
\][/tex]
Therefore, the solution to the equation [tex]\(2 x^{\frac{3}{2}} = 16\)[/tex] is:
[tex]\[
x = 4
\][/tex]
