Given the expressions [tex]\(\sqrt{60}\)[/tex] and [tex]\(\sqrt{114}\)[/tex], we want to find their simplified forms.
1. Simplifying [tex]\(\sqrt{60}\)[/tex]:
- The number 60 can be factored into prime factors: [tex]\(60 = 2^2 \cdot 3 \cdot 5\)[/tex].
- Using the property of square roots, we can separate the factors: [tex]\(\sqrt{60} = \sqrt{2^2 \cdot 3 \cdot 5}\)[/tex].
- We know that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], so we can write: [tex]\(\sqrt{60} = \sqrt{2^2} \cdot \sqrt{3 \cdot 5}\)[/tex].
- Simplifying further, [tex]\(\sqrt{2^2} = 2\)[/tex], so [tex]\(\sqrt{60} = 2 \cdot \sqrt{15}\)[/tex].
Therefore, [tex]\(\sqrt{60} = 2\sqrt{15}\)[/tex].
2. Simplifying [tex]\(\sqrt{114}\)[/tex]:
- The number 114 can be factored into prime factors: [tex]\(114 = 2 \cdot 3 \cdot 19\)[/tex].
- Using the property of square roots, we can separate the factors: [tex]\(\sqrt{114} = \sqrt{2 \cdot 3 \cdot 19}\)[/tex].
- Each of the factors (2, 3, and 19) is a prime number and cannot be simplified further under the square root.
Therefore, [tex]\(\sqrt{114}\)[/tex] remains as [tex]\(\sqrt{114}\)[/tex].
In conclusion, the simplified forms of the given expressions are:
[tex]\[
\sqrt{60} = 2\sqrt{15}
\][/tex]
[tex]\[
\sqrt{114} = \sqrt{114}
\][/tex]
