What is 1+ the square root of 2 over square root of 5. Find the denominator

The expression you're asking about is:

\[
1 + \frac{\sqrt{2}}{\sqrt{5}}
\]

To find the denominator, we should first simplify the fraction by rationalizing it.

### Step 1: Rationalize the denominator
The given expression is:

\[
\frac{\sqrt{2}}{\sqrt{5}}
\]

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{5}\):

\[
\frac{\sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{2} \times \sqrt{5}}{5} = \frac{\sqrt{10}}{5}
\]

### Step 2: Rewrite the original expression
Now, substitute back into the original expression:

\[
1 + \frac{\sqrt{10}}{5}
\]

### Step 3: Express with a common denominator
To express the entire expression as a single fraction with a common denominator, rewrite 1 as \(\frac{5}{5}\):

\[
\frac{5}{5} + \frac{\sqrt{10}}{5} = \frac{5 + \sqrt{10}}{5}
\]

### Denominator
The denominator of this simplified expression is 5.

Would you like more details or have any questions?

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Here are some related questions for further practice:

1. How do you rationalize a denominator in a fraction?
2. What is the value of \(\sqrt{10}\) approximately?
3. How do you add fractions with different denominators?
4. What is the process to simplify complex fractions?
5. How do you express a whole number as a fraction?

**Tip:** When rationalizing a denominator, always multiply by the same square root present in the denominator to remove it.

Learn how to simplify the expression 1 + sqrt(2) / sqrt(5) and find its denominator step-by-step. This problem involves rationalizing the denominator and adding fractions with different roots. Suitable for high school students.

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