To simplify the algebraic expression [tex]\(\sqrt{45 x^5}\)[/tex], follow these steps:
1. Factorize the coefficients:
- The number 45 can be factorized into prime factors as [tex]\(45 = 9 \times 5 = 3^2 \times 5\)[/tex].
2. Separate the variable part:
- The term [tex]\(x^5\)[/tex] can be rewritten as [tex]\(x^4 \times x\)[/tex], where [tex]\(x^4\)[/tex] is a perfect square.
3. Apply the square root to each part:
[tex]\[
\sqrt{45 x^5} = \sqrt{3^2 \times 5 \times x^4 \times x}
\][/tex]
Separate the square root into individual parts:
[tex]\[
\sqrt{3^2} \times \sqrt{5} \times \sqrt{x^4} \times \sqrt{x}
\][/tex]
4. Simplify each square root:
- [tex]\(\sqrt{3^2} = 3\)[/tex]
- [tex]\(\sqrt{5}\)[/tex] remains [tex]\(\sqrt{5}\)[/tex] since 5 is not a perfect square.
- [tex]\(\sqrt{x^4} = x^2\)[/tex]
- [tex]\(\sqrt{x}\)[/tex] remains [tex]\(\sqrt{x}\)[/tex]
5. Combine the simplified parts:
[tex]\[
\sqrt{45 x^5} = 3 \times \sqrt{5} \times x^2 \times \sqrt{x}
\][/tex]
6. Rearrange to form a final expression:
[tex]\[
= 3 x^2 \sqrt{5 x}
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt{45 x^5}\)[/tex] is [tex]\(\boxed{3 x^2 \sqrt{5 x}}\)[/tex].
The correct answer is:
B. [tex]\(3 x^2 \sqrt{5 x}\)[/tex]
