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Solve the following equation with a rational exponent:

[tex]\[2 x^{\frac{3}{2}} = 16\][/tex]

A. 8
B. 84
C. 4
D. no solution


Sagot :

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].