To simplify the radical expression [tex]\(\sqrt{250}\)[/tex], we need to find its prime factorization and look for perfect square factors. Here is the step-by-step process:
1. Prime Factorization:
- Start by determining the factors of 250.
- Notice that 250 can be factored as [tex]\(250 = 25 \times 10\)[/tex].
- Both 25 and 10 can be further factored into primes:
- [tex]\(25 = 5 \times 5\)[/tex]
- [tex]\(10 = 2 \times 5\)[/tex]
- So, the prime factorization of 250 is [tex]\(250 = 2 \times 5^3\)[/tex].
2. Identify Perfect Squares:
- Look at the prime factorization [tex]\(2 \times 5^3\)[/tex].
- Recognize that [tex]\(5^2\)[/tex] (which is 25) is a perfect square.
3. Rewrite the Radical:
- Rewrite [tex]\(\sqrt{250}\)[/tex] using its prime factors:
[tex]\[
\sqrt{250} = \sqrt{25 \times 10}
\][/tex]
- Since [tex]\(25\)[/tex] is a perfect square and [tex]\(\sqrt{25} = 5\)[/tex], we can break it down as follows:
[tex]\[
\sqrt{250} = \sqrt{25 \times 10} = \sqrt{25} \times \sqrt{10} = 5 \times \sqrt{10}
\][/tex]
4. Simplify Further:
- The simplified form of [tex]\(\sqrt{250}\)[/tex] is [tex]\(5 \sqrt{10}\)[/tex].
5. Choose the Correct Option:
- Based on the options given:
- (A) [tex]\(6 \sqrt{2}\)[/tex]
- (B) [tex]\(5 \sqrt{10}\)[/tex]
- (C) [tex]\(3 \sqrt{2}\)[/tex]
- (D) [tex]\(8 \sqrt{5}\)[/tex]
- (E) 50
The correct answer is option (B) [tex]\(5 \sqrt{10}\)[/tex].
Therefore, through this step-by-step simplification process, we find that [tex]\(\sqrt{250} = 5 \sqrt{10}\)[/tex], and the correct option is (B) [tex]\(5 \sqrt{10}\)[/tex].
