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].
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].
