Evaluate `sqrt(20)+sqrt(12)+root(3)(729)-(4)/(sqrt(5)-sqrt(3))-sqrt(81)`

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. `0`
D. `2sqrt(2)`

Answer

To evaluate the expression \( \sqrt{20} + \sqrt{12} + \sqrt[3]{729} - \frac{4}{\sqrt{5} - \sqrt{3}} - \sqrt{81} \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{20} \)
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]

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

### Step 2: Simplify \( \sqrt{12} \)
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

**Hint:** Look for pairs of factors that are perfect squares.

### Step 3: Simplify \( \sqrt[3]{729} \)
\[ \sqrt[3]{729} = 9 \quad \text{(since } 9 \times 9 \times 9 = 729\text{)} \]

**Hint:** Recognize that \( 729 \) is a power of \( 9 \) (specifically, \( 9^3 \)).

### Step 4: Simplify \( \sqrt{81} \)
\[ \sqrt{81} = 9 \]

**Hint:** Identify perfect squares directly.

### Step 5: Substitute the simplified values into the expression
Now substituting the simplified values back into the expression:
\[ 2\sqrt{5} + 2\sqrt{3} + 9 - \frac{4}{\sqrt{5} - \sqrt{3}} - 9 \]

### Step 6: Combine like terms
The \( 9 \) and \( -9 \) cancel out:
\[ 2\sqrt{5} + 2\sqrt{3} - \frac{4}{\sqrt{5} - \sqrt{3}} \]

### Step 7: Rationalize the denominator of \( \frac{4}{\sqrt{5} - \sqrt{3}} \)
Multiply the numerator and denominator by \( \sqrt{5} + \sqrt{3} \):
\[ \frac{4(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} = \frac{4(\sqrt{5} + \sqrt{3})}{5 - 3} = \frac{4(\sqrt{5} + \sqrt{3})}{2} = 2(\sqrt{5} + \sqrt{3}) \]

**Hint:** To rationalize, multiply by the conjugate of the denominator.

### Step 8: Substitute back into the expression
Now we have:
\[ 2\sqrt{5} + 2\sqrt{3} - 2(\sqrt{5} + \sqrt{3}) \]

### Step 9: Combine like terms again
\[ 2\sqrt{5} + 2\sqrt{3} - 2\sqrt{5} - 2\sqrt{3} = 0 \]

### Final Result
The final result of the expression is:
\[ \boxed{0} \]