To simplify the expression [tex]\( 6 \sqrt{5} (\sqrt{8} + \sqrt{18}) \)[/tex] and express it in the form [tex]\( a \sqrt{10} \)[/tex], let's go through the steps systematically.
1. Simplifying [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2}
\][/tex]
2. Simplifying [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}
\][/tex]
3. Adding the simplified square roots:
[tex]\[
\sqrt{8} + \sqrt{18} = 2 \sqrt{2} + 3 \sqrt{2} = (2 + 3) \sqrt{2} = 5 \sqrt{2}
\][/tex]
4. Multiplying by [tex]\( 6 \sqrt{5} \)[/tex]:
[tex]\[
6 \sqrt{5} \cdot 5 \sqrt{2}
\][/tex]
Combine the constants and the square roots separately:
[tex]\[
6 \cdot 5 \cdot \sqrt{5} \cdot \sqrt{2} = 30 \cdot \sqrt{10}
\][/tex]
5. Therefore, the expression [tex]\( 6 \sqrt{5} (\sqrt{8} + \sqrt{18}) \)[/tex] can be written as [tex]\( 30 \sqrt{10} \)[/tex].
We can now compare this with [tex]\( a \sqrt{10} \)[/tex] and see that [tex]\( a = 30 \)[/tex].
Thus, the value of [tex]\( a \)[/tex] is:
[tex]\[
a = 30
\][/tex]
