To solve the equation [tex]\( 2(x-4)^{\frac{3}{2}} = 54 \)[/tex], follow these steps:
1. Isolate the power term: First, divide both sides of the equation by 2 in order to isolate [tex]\( (x-4)^{\frac{3}{2}} \)[/tex].
[tex]\[
\frac{2}{2}(x-4)^{\frac{3}{2}} = \frac{54}{2}
\][/tex]
Simplifying this, we get:
[tex]\[
(x-4)^{\frac{3}{2}} = 27
\][/tex]
2. Eliminate the fractional exponent: To eliminate the [tex]\(\frac{3}{2}\)[/tex] exponent, raise both sides of the equation to the power of [tex]\(\frac{2}{3}\)[/tex]. This will effectively reverse the exponent operation.
[tex]\[
\left[(x-4)^{\frac{3}{2}}\right]^{\frac{2}{3}} = 27^{\frac{2}{3}}
\][/tex]
3. Simplify both sides: Since [tex]\(\left(a^{m}\right)^{n} = a^{mn}\)[/tex], the left-hand side simplifies to [tex]\( x - 4 \)[/tex].
[tex]\[
x - 4 = 27^{\frac{2}{3}}
\][/tex]
Now evaluate [tex]\( 27^{\frac{2}{3}} \)[/tex]:
- First, calculate the cube root of 27, which is 3 (since [tex]\( 3^3 = 27 \)[/tex]).
- Then raise the result to the power of 2:
[tex]\[
27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2 = 3^2 = 9
\][/tex]
Therefore:
[tex]\[
x - 4 = 9
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Finally, add 4 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 9 + 4
\][/tex]
Simplifying this, we get:
[tex]\[
x = 13
\][/tex]
Thus, the solution to the equation [tex]\( 2(x-4)^{\frac{3}{2}} = 54 \)[/tex] is [tex]\( x = 13 \)[/tex].
The correct answer is:
B. 13
