Certainly! Let's work out the expression [tex]\(\frac{\sqrt{3}}{\sqrt{10}} + \frac{3 \sqrt{6}}{\sqrt{5}}\)[/tex] and simplify it to the form [tex]\(\frac{c \sqrt{d}}{10}\)[/tex], where [tex]\(c\)[/tex] and [tex]\(d\)[/tex] are integers.
First, let's simplify each term of the expression separately:
### Simplification of [tex]\(\frac{\sqrt{3}}{\sqrt{10}}\)[/tex]
We start by combining the square roots:
[tex]\[
\frac{\sqrt{3}}{\sqrt{10}} = \sqrt{\frac{3}{10}}
\][/tex]
We can simplify the fraction inside the square root:
[tex]\[
\sqrt{\frac{3}{10}} = \frac{\sqrt{3 \times 1}}{\sqrt{10}} = \frac{\sqrt{30}}{10}
\][/tex]
So, we have:
[tex]\[
\frac{\sqrt{3}}{\sqrt{10}} = \frac{\sqrt{30}}{10}
\][/tex]
### Simplification of [tex]\(\frac{3 \sqrt{6}}{\sqrt{5}}\)[/tex]
Next, let's simplify this term:
[tex]\[
\frac{3 \sqrt{6}}{\sqrt{5}} = 3 \cdot \frac{\sqrt{6}}{\sqrt{5}} = 3 \cdot \sqrt{\frac{6}{5}}
\][/tex]
We can simplify the fraction inside the square root:
[tex]\[
\sqrt{\frac{6}{5}} = \frac{\sqrt{6 \times 1}}{\sqrt{5}} = \frac{\sqrt{30}}{5}
\][/tex]
Thus, we can write:
[tex]\[
3 \cdot \sqrt{\frac{6}{5}} = 3 \cdot \frac{\sqrt{30}}{5} = \frac{3 \sqrt{30}}{5}
\][/tex]
Now, recognizing that [tex]\(\frac{3}{5}\)[/tex] as a multiplication factor:
[tex]\[
\frac{3 \sqrt{30}}{5} = \frac{6 \sqrt{30}}{10}
\][/tex]
### Combining the Terms
Now we combine both simplified terms:
[tex]\[
\frac{\sqrt{3}}{\sqrt{10}} + \frac{3 \sqrt{6}}{\sqrt{5}} = \frac{\sqrt{30}}{10} + \frac{6 \sqrt{30}}{10}
\][/tex]
Since both terms have the same denominator, we can combine them:
[tex]\[
\frac{\sqrt{30}}{10} + \frac{6 \sqrt{30}}{10} = \frac{(1+6)\sqrt{30}}{10} = \frac{7 \sqrt{30}}{10}
\][/tex]
### Final Form
Thus, the expression [tex]\(\frac{\sqrt{3}}{\sqrt{10}} + \frac{3 \sqrt{6}}{\sqrt{5}}\)[/tex] can be written in the form [tex]\(\frac{c \sqrt{d}}{10}\)[/tex] where:
[tex]\[
c = 7 \quad \text{and} \quad d = 30
\][/tex]
So, the values of [tex]\(c\)[/tex] and [tex]\(d\)[/tex] are:
[tex]\[
c = 7, \quad d = 30
\][/tex]
