If a, b are rational numbers and `asqrtbsqrt3=sqrt(98)+sqrt(108)-sqrt(48)-sqrt(72)`, then the values of a and b are?

Options

A. 1,2
B. 1,3
C. 2,1
D. 2,3

Answer

To solve the equation \( a \sqrt{b} \sqrt{3} = \sqrt{98} + \sqrt{108} - \sqrt{48} - \sqrt{72} \), we will simplify the right-hand side step by step and then compare coefficients to find the values of \( a \) and \( b \).

### Step 1: Simplify \( \sqrt{98} \)
\[ \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2} \]

**Hint:** Factor the number under the square root to find perfect squares.

### Step 2: Simplify \( \sqrt{108} \)
\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \]

**Hint:** Look for the largest perfect square that divides the number.

### Step 3: Simplify \( \sqrt{48} \)
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \]

**Hint:** Again, factor to find perfect squares.

### Step 4: Simplify \( \sqrt{72} \)
\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \]

**Hint:** Use the same method of factoring to simplify.

### Step 5: Combine the simplified terms
Now substitute back into the equation:
\[ \sqrt{98} + \sqrt{108} - \sqrt{48} - \sqrt{72} = 7\sqrt{2} + 6\sqrt{3} - 4\sqrt{3} - 6\sqrt{2} \]

Combine like terms:
\[ (7\sqrt{2} - 6\sqrt{2}) + (6\sqrt{3} - 4\sqrt{3}) = 1\sqrt{2} + 2\sqrt{3} \]

### Step 6: Set the equation equal to \( a \sqrt{b} \sqrt{3} \)
Now we have:
\[ a \sqrt{b} \sqrt{3} = 1\sqrt{2} + 2\sqrt{3} \]

### Step 7: Compare coefficients
From the equation \( a \sqrt{b} \sqrt{3} = 1\sqrt{2} + 2\sqrt{3} \), we can see that:
- The coefficient of \( \sqrt{2} \) is \( 1 \)
- The coefficient of \( \sqrt{3} \) is \( 2 \)

This implies:
- \( a \sqrt{b} = 2 \) (coefficient of \( \sqrt{3} \))
- \( a \) must be \( 1 \) (coefficient of \( \sqrt{2} \))

### Step 8: Solve for \( a \) and \( b \)
From \( a = 1 \):
\[ 1 \cdot \sqrt{b} = 2 \Rightarrow \sqrt{b} = 2 \Rightarrow b = 4 \]

### Final Values
Thus, the values of \( a \) and \( b \) are:
\[ a = 1, \quad b = 4 \]

### Summary
The final answer is:
- \( a = 1 \)
- \( b = 4 \)