To solve the equation [tex]\(2 x^{\frac{3}{2}} = 16\)[/tex], let's go through the steps methodically:
1. Isolate [tex]\(x^{\frac{3}{2}}\)[/tex]:
[tex]\[
\frac{2 x^{\frac{3}{2}}}{2} = \frac{16}{2}
\][/tex]
[tex]\[
x^{\frac{3}{2}} = 8
\][/tex]
2. Solve for [tex]\( x \)[/tex] by raising both sides of the equation to the power of [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[
\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}
\][/tex]
Applying the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[
x^{( \frac{3}{2} \cdot \frac{2}{3} )} = 8^{\frac{2}{3}}
\][/tex]
Simplifying the exponent:
[tex]\[
x^1 = 8^{\frac{2}{3}}
\][/tex]
Thus:
[tex]\[
x = 8^{\frac{2}{3}}
\][/tex]
3. Calculate [tex]\( 8^{\frac{2}{3}} \)[/tex]:
We know that:
[tex]\[
8 = 2^3
\][/tex]
So:
[tex]\[
8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}}
\][/tex]
Applying the property of exponents:
[tex]\[
(2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4
\][/tex]
4. Confirming the solution:
Substitute [tex]\(x = 4\)[/tex] back into the original equation to check the validity:
[tex]\[
2 \cdot (4)^{\frac{3}{2}}
\][/tex]
Calculate [tex]\(4^{\frac{3}{2}}\)[/tex]:
[tex]\[
4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^{2 \cdot \frac{3}{2}} = 2^3 = 8
\][/tex]
Therefore:
[tex]\[
2 \cdot 8 = 16
\][/tex]
The left-hand side equals the right-hand side confirming our solution.
Hence, the solution to the equation [tex]\(2 x^{\frac{3}{2}} = 16\)[/tex] is [tex]\(x = 4\)[/tex].
