Simplify : ` ((2 sqrt(45) + 3 sqrt(20)))/(2sqrt5)`

Options

A. 2
B. 4
C. 8
D. 6

Answer

To simplify the expression \(\frac{2 \sqrt{45} + 3 \sqrt{20}}{2 \sqrt{5}}\), we will follow these steps:

### Step 1: Simplify \(\sqrt{45}\) and \(\sqrt{20}\)
First, we will simplify \(\sqrt{45}\) and \(\sqrt{20}\):
- \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\)
- \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\)

### Step 2: Substitute the simplified square roots back into the expression
Now, we substitute these simplified forms back into the original expression:
\[ 2 \sqrt{45} + 3 \sqrt{20} = 2(3\sqrt{5}) + 3(2\sqrt{5}) = 6\sqrt{5} + 6\sqrt{5} \]

### Step 3: Combine like terms
Next, we combine the like terms:
\[ 6\sqrt{5} + 6\sqrt{5} = 12\sqrt{5} \]

### Step 4: Rewrite the expression
Now, we rewrite the entire expression:
\[ \frac{12\sqrt{5}}{2\sqrt{5}} \]

### Step 5: Simplify the fraction
Now, we can simplify the fraction:
\[ \frac{12\sqrt{5}}{2\sqrt{5}} = \frac{12}{2} = 6 \]

### Final Answer
Thus, the simplified answer is:
\[ 6 \]

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