If `(sqrt((1058) xx (648)))/(x) = 18`, then the value of x is

Options

A. 36
B. 46
C. 58
D. 23

Answer

To solve the equation \(\frac{\sqrt{(1058) \times (648)}}{x} = 18\), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation to isolate the square root term:
\[ \sqrt{(1058) \times (648)} = 18x \]

### Step 2: Calculate the product inside the square root
Now, we need to calculate \(1058 \times 648\). However, we will simplify it using prime factorization:
\[ 1058 = 2 \times 529 \quad \text{(since } 529 = 23^2\text{)} \]
\[ 648 = 2^3 \times 81 = 2^3 \times 3^4 \]
Thus:
\[ 1058 \times 648 = (2 \times 529) \times (2^3 \times 3^4) = 2^4 \times 529 \times 3^4 \]

### Step 3: Substitute back into the square root
Now we substitute this back into our equation:
\[ \sqrt{(2^4) \times (23^2) \times (3^4)} \]

### Step 4: Simplify the square root
Using the property of square roots, we can simplify:
\[ \sqrt{2^4} = 2^2 = 4, \quad \sqrt{23^2} = 23, \quad \sqrt{3^4} = 3^2 = 9 \]
Thus:
\[ \sqrt{(2^4) \times (23^2) \times (3^4)} = 4 \times 23 \times 9 \]

### Step 5: Calculate the product
Now we calculate:
\[ 4 \times 23 = 92 \]
\[ 92 \times 9 = 828 \]
So, we have:
\[ \sqrt{(1058) \times (648)} = 828 \]

### Step 6: Substitute back into the equation
Now we substitute back into our equation:
\[ 828 = 18x \]

### Step 7: Solve for x
Now, divide both sides by 18 to find \(x\):
\[ x = \frac{828}{18} \]

### Step 8: Simplify the division
Calculating the division:
\[ x = 46 \]

Thus, the value of \(x\) is:
\[ \boxed{46} \]

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