Express as an equivalent fraction with rational denominator.
`([sqrt(45)-sqrt(20)][sqrt(12)+sqrt(75)]sqrt(3))/(sqrt((5))+sqrt((180)))`

Answer

To express the given expression as an equivalent fraction with a rational denominator, we will simplify the expression step by step.

**Given Expression:**
\[ \frac{(\sqrt{45} - \sqrt{20})(\sqrt{12} + \sqrt{75})\sqrt{3}}{\sqrt{5} + \sqrt{180}} \]

### Step 1: Simplify the square roots in the numerator

1. **Simplify \(\sqrt{45}\):**
\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]

2. **Simplify \(\sqrt{20}\):**
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]

3. **Now substitute back into the expression:**
\[ \sqrt{45} - \sqrt{20} = 3\sqrt{5} - 2\sqrt{5} = (3 - 2)\sqrt{5} = \sqrt{5} \]

4. **Simplify \(\sqrt{12}\):**
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]

5. **Simplify \(\sqrt{75}\):**
\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \]

6. **Now substitute back into the expression:**
\[ \sqrt{12} + \sqrt{75} = 2\sqrt{3} + 5\sqrt{3} = (2 + 5)\sqrt{3} = 7\sqrt{3} \]

### Step 2: Substitute simplified values into the numerator

The numerator now becomes:
\[ \sqrt{5} \cdot 7\sqrt{3} \cdot \sqrt{3} = 7\sqrt{5} \cdot 3 = 21\sqrt{5} \]

### Step 3: Simplify the denominator

1. **Simplify \(\sqrt{180}\):**
\[ \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5} \]

2. **Now substitute back into the expression:**
\[ \sqrt{5} + \sqrt{180} = \sqrt{5} + 6\sqrt{5} = (1 + 6)\sqrt{5} = 7\sqrt{5} \]

### Step 4: Substitute simplified values into the denominator

The denominator now becomes:
\[ 7\sqrt{5} \]

### Step 5: Combine the simplified numerator and denominator

Now we can rewrite the entire expression:
\[ \frac{21\sqrt{5}}{7\sqrt{5}} \]

### Step 6: Simplify the fraction

1. **Cancel \(\sqrt{5}\):**
\[ \frac{21}{7} = 3 \]

Thus, the expression simplifies to:
\[ \boxed{3} \]

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