Find the smallest rationalizing factor is `sqrt(28)` .

Answer

To find the smallest rationalizing factor of \(\sqrt{28}\), we can follow these steps:

### Step 1: Factor 28
First, we need to factor the number 28 into its prime factors.

\[ 28 = 2 \times 14 \]
\[ 14 = 2 \times 7 \]

So, we can write:
\[ 28 = 2^2 \times 7 \]

### Step 2: Simplify \(\sqrt{28}\)
Next, we simplify \(\sqrt{28}\) using the prime factorization we found.

\[ \sqrt{28} = \sqrt{2^2 \times 7} \]

Using the property of square roots, we can separate the factors:

\[ \sqrt{28} = \sqrt{2^2} \times \sqrt{7} \]

Since \(\sqrt{2^2} = 2\), we have:

\[ \sqrt{28} = 2\sqrt{7} \]

### Step 3: Identify the Rationalizing Factor
To rationalize \(\sqrt{28}\), we need to eliminate the square root from the denominator if it appears in a fraction. In this case, we need to find the smallest rationalizing factor that will make \(\sqrt{7}\) a whole number.

The rationalizing factor for \(\sqrt{7}\) is \(\sqrt{7}\) itself. Therefore, we multiply by \(\sqrt{7}\):

\[ \text{Rationalizing factor} = \sqrt{7} \]

### Final Answer
Thus, the smallest rationalizing factor of \(\sqrt{28}\) is:

\[ \sqrt{7} \]

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