Write the lowest rationalising factor of :
`sqrt(18)-sqrt(50)`

Answer

To find the lowest rationalizing factor of the expression \( \sqrt{18} - \sqrt{50} \), we will follow these steps:

### Step 1: Simplify \( \sqrt{18} \) and \( \sqrt{50} \)

1. **Simplifying \( \sqrt{18} \)**:
- The prime factorization of 18 is \( 2 \times 3^2 \).
- Therefore, \( \sqrt{18} = \sqrt{2 \times 3^2} = \sqrt{3^2} \times \sqrt{2} = 3\sqrt{2} \).

2. **Simplifying \( \sqrt{50} \)**:
- The prime factorization of 50 is \( 2 \times 5^2 \).
- Therefore, \( \sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{5^2} \times \sqrt{2} = 5\sqrt{2} \).

### Step 2: Rewrite the expression

Now we can rewrite the expression \( \sqrt{18} - \sqrt{50} \) using the simplified forms:
\[ \sqrt{18} - \sqrt{50} = 3\sqrt{2} - 5\sqrt{2} \]

### Step 3: Combine like terms

Since both terms contain \( \sqrt{2} \), we can combine them:
\[ 3\sqrt{2} - 5\sqrt{2} = (3 - 5)\sqrt{2} = -2\sqrt{2} \]

### Step 4: Identify the lowest rationalizing factor

The expression \( -2\sqrt{2} \) contains the irrational number \( \sqrt{2} \). To rationalize it, we need to multiply by \( \sqrt{2} \). Therefore, the lowest rationalizing factor is:
\[ \sqrt{2} \]

### Final Answer

The lowest rationalizing factor of \( \sqrt{18} - \sqrt{50} \) is \( \sqrt{2} \).

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