By how much does `(sqrt(12)+sqrt(18))` exceed `(2sqrt(3)+2sqrt(2))` ?

Options

A. `2`
B. `sqrt(3)`
C. `sqrt(2)`
D. `3`

Answer

To solve the problem of how much \((\sqrt{12} + \sqrt{18})\) exceeds \((2\sqrt{3} + 2\sqrt{2})\), we will follow these steps:

### Step 1: Simplify \(\sqrt{12}\) and \(\sqrt{18}\)

We start by simplifying \(\sqrt{12}\) and \(\sqrt{18}\).

\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]

### Step 2: Substitute back into the expression

Now we substitute these simplified forms back into the original expression:

\[ \sqrt{12} + \sqrt{18} = 2\sqrt{3} + 3\sqrt{2} \]

### Step 3: Set up the inequality

Next, we need to find out how much \((\sqrt{12} + \sqrt{18})\) exceeds \((2\sqrt{3} + 2\sqrt{2})\). We will set up the expression:

\[ (2\sqrt{3} + 3\sqrt{2}) - (2\sqrt{3} + 2\sqrt{2}) \]

### Step 4: Simplify the expression

Now we simplify the expression:

\[ (2\sqrt{3} + 3\sqrt{2}) - (2\sqrt{3} + 2\sqrt{2}) = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]

### Conclusion

Thus, \((\sqrt{12} + \sqrt{18})\) exceeds \((2\sqrt{3} + 2\sqrt{2})\) by \(\sqrt{2}\).

### Final Answer

The answer is \(\sqrt{2}\).

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