To solve the problem of simplifying the expression [tex]\(\sqrt{150}\)[/tex], we need to break down the number 150 into its prime factors.
1. First, we factorize 150:
[tex]\[ 150 = 2 \times 3 \times 5^2 \][/tex]
2. Next, we apply the property of square roots that states [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{150} = \sqrt{2 \times 3 \times 25} \][/tex]
3. We know that the square root of a product of numbers is equal to the product of the square roots of the individual numbers:
[tex]\[ \sqrt{2 \times 3 \times 25} = \sqrt{2} \times \sqrt{3} \times \sqrt{25} \][/tex]
4. We know the value of [tex]\(\sqrt{25}\)[/tex] is 5:
[tex]\[ \sqrt{2} \times \sqrt{3} \times \sqrt{25} = 5 \times \sqrt{2 \times 3} = 5 \times \sqrt{6}\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{150}\)[/tex] is [tex]\(5 \sqrt{6}\)[/tex].
Among the given choices:
- [tex]\(25 \sqrt{6}\)[/tex]
- [tex]\(25 \sqrt{3}\)[/tex]
- [tex]\(15 \sqrt{10}\)[/tex]
- [tex]\(5 \sqrt{6}\)[/tex]
The correct answer is:
[tex]\(5 \sqrt{6}\)[/tex]
