`sqrt225xx(12)^2 + 37=(?)^3`

Options

A. 64
B. 13
C. `sqrt6`
D. 6

Answer

To solve the equation \( \sqrt{225 \times (12)^2} + 37 = (?)^3 \), we will follow these steps:

### Step 1: Simplify the square root
First, we simplify the expression inside the square root:
\[ \sqrt{225 \times (12)^2} \]

### Step 2: Calculate \( (12)^2 \)
Calculating \( (12)^2 \):
\[ (12)^2 = 144 \]

### Step 3: Multiply 225 by 144
Now we multiply 225 by 144:
\[ 225 \times 144 \]

### Step 4: Factor 225
We can express 225 as \( 15^2 \):
\[ 225 = 15^2 \]
Thus,
\[ 225 \times 144 = 15^2 \times 144 \]

### Step 5: Calculate \( 15^2 \times 144 \)
Now we can calculate:
\[ 15^2 = 225 \quad \text{and} \quad 144 = 12^2 \]
So,
\[ 225 \times 144 = 15^2 \times 12^2 = (15 \times 12)^2 = 180^2 \]

### Step 6: Take the square root
Taking the square root:
\[ \sqrt{225 \times 144} = 180 \]

### Step 7: Add 37
Now we add 37 to the result:
\[ 180 + 37 = 217 \]

### Step 8: Set up the equation
Now we have:
\[ 217 = (?)^3 \]

### Step 9: Find the cube root
To find \( ? \), we need to compute the cube root of 217:
\[ ? = \sqrt[3]{217} \]

### Step 10: Calculate the cube root
We can estimate the cube root of 217. We know that:
\[ 6^3 = 216 \quad \text{and} \quad 7^3 = 343 \]
Thus, \( ? \) is approximately 6.

### Final Answer
Since \( 6^3 = 216 \) and \( 7^3 = 343 \), we conclude that:
\[ ? = 6 \]