Answer :

To solve the equation \(2(x-4)^{\frac{3}{2}}=54\), we will go through it step-by-step:

1. Isolate the expression involving \(x\)

[tex]\[ 2(x-4)^{\frac{3}{2}} = 54 \][/tex]

Divide both sides by 2 to simplify:

[tex]\[ (x-4)^{\frac{3}{2}} = \frac{54}{2} = 27 \][/tex]

2. Solve for \(x-4\)

To isolate \( x - 4 \), we need to get rid of the exponent \( \frac{3}{2} \). We do this by raising both sides of the equation to the power of \( \frac{2}{3} \):

[tex]\[ (x-4) = 27^{\frac{2}{3}} \][/tex]

3. Simplify \( 27^{\frac{2}{3}} \)

Recall that 27 can be expressed as \(3^3\):

[tex]\[ 27 = 3^3 \][/tex]

Thus,

[tex]\[ 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9 \][/tex]

Hence,

[tex]\[ x - 4 = 9 \][/tex]

4. Solve for \(x\)

Add 4 to both sides:

[tex]\[ x = 9 + 4 = 13 \][/tex]

Therefore, the solution to the equation \(2(x-4)^{\frac{3}{2}}=54\) is:

[tex]\[ \boxed{13} \][/tex]

So, the correct answer is [tex]\( \text{B. 13} \)[/tex].