Find the positive square root of `14sqrt(5)-30`

Options

A. `sqrt(5)(3-sqrt(5))`
B. `root4(5)(3-sqrt(5))`
C. `sqrt(3)+2sqrt(5)`
D. `sqrt(3)-2sqrt(5)`

Answer

To find the positive square root of \( 14\sqrt{5} - 30 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( 14\sqrt{5} - 30 \).

### Step 2: Factor out common terms
Notice that \( 30 \) can be expressed in terms of \( \sqrt{5} \). We can rewrite \( 30 \) as \( 2 \times 3 \times 5 \). Thus, we have:
\[ 14\sqrt{5} - 30 = 14\sqrt{5} - 2 \times 3 \times 5 \]

### Step 3: Factor out \( \sqrt{5} \)
Now, we can factor out \( \sqrt{5} \):
\[ = \sqrt{5}(14 - 2 \times 3) \]
Calculating \( 2 \times 3 = 6 \), we have:
\[ = \sqrt{5}(14 - 6) = \sqrt{5}(8) \]

### Step 4: Recognize a perfect square
Next, we can express \( 14\sqrt{5} - 30 \) as a perfect square. We can rewrite \( 14\sqrt{5} - 30 \) in the form of a square:
\[ = \sqrt{5} \left( 3 - \sqrt{5} \right)^2 \]
This is because:
\[ (3 - \sqrt{5})^2 = 3^2 - 2 \times 3 \times \sqrt{5} + (\sqrt{5})^2 = 9 - 6\sqrt{5} + 5 = 14 - 6\sqrt{5} \]

### Step 5: Take the square root
Now we take the square root of the entire expression:
\[ \sqrt{14\sqrt{5} - 30} = \sqrt{\sqrt{5} (3 - \sqrt{5})^2} \]
This simplifies to:
\[ = \sqrt{5} \cdot (3 - \sqrt{5}) \]

### Step 6: Final expression
Thus, the positive square root of \( 14\sqrt{5} - 30 \) is:
\[ \sqrt{5}(3 - \sqrt{5}) \]