Let's break down the expression [tex]\(\sqrt{5} + \sqrt{125} + 7 \sqrt{5}\)[/tex] step by step.
1. Identify and simplify the square roots:
First, let's simplify [tex]\(\sqrt{125}\)[/tex].
[tex]\[
\sqrt{125} = \sqrt{5 \times 25} = \sqrt{5} \times \sqrt{25} = \sqrt{5} \times 5 = 5\sqrt{5}
\][/tex]
So, the expression becomes:
[tex]\[
\sqrt{5} + 5\sqrt{5} + 7\sqrt{5}
\][/tex]
2. Combine like terms:
Notice that all the terms have [tex]\(\sqrt{5}\)[/tex] as a common factor:
[tex]\[
\sqrt{5} + 5\sqrt{5} + 7\sqrt{5} = (1\sqrt{5} + 5\sqrt{5} + 7\sqrt{5})
\][/tex]
We can factor out [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
(1 + 5 + 7) \sqrt{5}
\][/tex]
Simplify inside the parentheses:
[tex]\[
1 + 5 + 7 = 13
\][/tex]
So, the expression becomes:
[tex]\[
13\sqrt{5}
\][/tex]
3. Calculate the numerical value:
We can now evaluate the numerical value of [tex]\(13\sqrt{5}\)[/tex]:
[tex]\[
\sqrt{5} \approx 2.23606797749979
\][/tex]
Thus,
[tex]\[
13 \times 2.23606797749979 \approx 29.068883707497267
\][/tex]
Therefore, the simplified expression [tex]\(\sqrt{5} + \sqrt{125} + 7 \sqrt{5}\)[/tex] equals:
[tex]\[
13\sqrt{5} \approx 29.068883707497267
\][/tex]
Each step logically leads us to the final numerical result that approximates to [tex]\(29.068883707497267\)[/tex].
