To simplify the expression [tex]\(\sqrt{150}\)[/tex] by factoring, we need to break down the number 150 into its prime factors and then simplify the square root.
### Step-by-Step Solution:
1. Factorizing [tex]\(150\)[/tex] into Prime Factors:
- Start with the smallest prime number [tex]\(2\)[/tex]:
[tex]\[
150 \div 2 = 75
\][/tex]
So, [tex]\(2\)[/tex] is a factor and we have [tex]\(75\)[/tex] left.
- Next, factor [tex]\(75\)[/tex] using the next smallest prime number [tex]\(3\)[/tex]:
[tex]\[
75 \div 3 = 25
\][/tex]
So, [tex]\(3\)[/tex] is a factor and we have [tex]\(25\)[/tex] left.
- Finally, factor [tex]\(25\)[/tex] using [tex]\(5\)[/tex]:
[tex]\[
25 \div 5 = 5
\][/tex]
and
[tex]\[
5 \div 5 = 1
\][/tex]
So, [tex]\(5\)[/tex] is a factor, and we are left with [tex]\(1\)[/tex], which means the number has been fully factorized.
2. Putting it All Together (Prime Factorization of [tex]\(150\)[/tex]):
[tex]\[
150 = 2 \times 3 \times 5 \times 5
\][/tex]
We can group the pairs of identical factors:
[tex]\[
150 = 2 \times 3 \times 5^2
\][/tex]
3. Simplifying the Square Root:
- The square root of a product is the product of the square roots:
[tex]\[
\sqrt{150} = \sqrt{2 \times 3 \times 5^2}
\][/tex]
- Use the property of square roots: [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{2 \times 3 \times 5^2} = \sqrt{2} \times \sqrt{3} \times \sqrt{5^2}
\][/tex]
- Since [tex]\(\sqrt{5^2} = 5\)[/tex]:
[tex]\[
\sqrt{2} \times \sqrt{3} \times 5
\][/tex]
4. Final Simplified Form:
[tex]\[
\sqrt{150} = 5 \sqrt{6}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{150}\)[/tex] is:
[tex]\[
\boxed{5\sqrt{6}}
\][/tex]
