To solve the expression [tex]\(\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} = a \sqrt{5} + b \sqrt{2}\)[/tex], let's follow the steps to identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
### Step 1: Rationalize the Denominator
First, we rationalize the denominator by multiplying both the numerator and the denominator by [tex]\(\sqrt{10}\)[/tex]:
[tex]\[
\frac{(\sqrt{2} + \sqrt{5}) \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}}
\][/tex]
### Step 2: Simplify the Expression
Let's simplify the expression step by step:
[tex]\[
\frac{(\sqrt{2} \cdot \sqrt{10}) + (\sqrt{5} \cdot \sqrt{10})}{10}
\][/tex]
[tex]\[
\sqrt{2} \cdot \sqrt{10} = \sqrt{2 \cdot 10} = \sqrt{20}
\][/tex]
[tex]\[
\sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50}
\][/tex]
So, the numerator simplifies to:
[tex]\[
\sqrt{20} + \sqrt{50}
\][/tex]
### Step 3: Further Simplify the Square Roots
Now, let's simplify [tex]\(\sqrt{20}\)[/tex] and [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\][/tex]
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{2\sqrt{5} + 5\sqrt{2}}{10}
\][/tex]
### Step 4: Distribute the Denominator
Now, distribute the denominator 10:
[tex]\[
\frac{2\sqrt{5}}{10} + \frac{5\sqrt{2}}{10}
\][/tex]
[tex]\[
\frac{2\sqrt{5}}{10} = \frac{2}{10} \cdot \sqrt{5} = \frac{1}{5}\sqrt{5}
\][/tex]
[tex]\[
\frac{5\sqrt{2}}{10} = \frac{5}{10} \cdot \sqrt{2} = \frac{1}{2}\sqrt{2}
\][/tex]
Combining these, we have:
[tex]\[
\frac{1}{5}\sqrt{5} + \frac{1}{2}\sqrt{2}
\][/tex]
### Step 5: Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
Comparing this with the original expression [tex]\(\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} = a \sqrt{5} + b \sqrt{2}\)[/tex], we get:
[tex]\[
a = \frac{1}{5} \quad \text{and} \quad b = \frac{1}{2}
\][/tex]
Therefore,
[tex]\[
\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} = \frac{1}{5}\sqrt{5} + \frac{1}{2}\sqrt{2}
\][/tex]
The values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are [tex]\(a = 0.2\)[/tex] and [tex]\(b = 0.5\)[/tex].
We also have the simplified expression:
[tex]\[
\frac{\sqrt{2} + \sqrt{5}}{\sqrt{10}} \approx 1.1543203766865053
\][/tex]
Finally, we have:
[tex]\[
a = 0.2 \quad \text{and} \quad b = 0.5
\][/tex]
