`((?)^(1.8))/(72) = (24)/((?)^(1.2))`

Options

A. 8
B. 14
C. 6
D. 12

Answer

To solve the equation \(\frac{(x)^{1.8}}{72} = \frac{24}{(x)^{1.2}}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions. This gives us:
\[ (x)^{1.8} \cdot (x)^{1.2} = 24 \cdot 72 \]

### Step 2: Combine Exponents
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\), we can combine the left side:
\[ (x)^{1.8 + 1.2} = 24 \cdot 72 \]
This simplifies to:
\[ (x)^{3.0} = 24 \cdot 72 \]

### Step 3: Calculate the Right Side
Now we calculate \(24 \cdot 72\):
\[ 24 \cdot 72 = 1728 \]
So, we have:
\[ (x)^3 = 1728 \]

### Step 4: Solve for \(x\)
To find \(x\), we take the cube root of both sides:
\[ x = \sqrt[3]{1728} \]

### Step 5: Calculate the Cube Root
Now we calculate the cube root of \(1728\):
\[ \sqrt[3]{1728} = 12 \]

Thus, the value of the question mark \(x\) is:
\[ \boxed{12} \]