To simplify the given expression [tex]\( \sqrt{28} \cdot \sqrt{27} \cdot \sqrt{5} \)[/tex], let's follow the step-by-step process:
1. Express the product of the square roots as a single square root:
[tex]\[
\sqrt{28} \cdot \sqrt{27} \cdot \sqrt{5} = \sqrt{28 \cdot 27 \cdot 5}
\][/tex]
2. Calculate the product under the square root:
[tex]\[
28 \cdot 27 \cdot 5 = 3780
\][/tex]
3. Simplify the square root of the product:
The expression becomes [tex]\( \sqrt{3780} \)[/tex].
4. Find the simplified form of [tex]\( \sqrt{3780} \)[/tex]:
- First, factorize 3780 into its prime factors:
[tex]\[
3780 = 2^2 \cdot 3^3 \cdot 5 \cdot 7
\][/tex]
- Pair the perfect squares outside the square root:
[tex]\[
\sqrt{2^2 \cdot 3^3 \cdot 5 \cdot 7} = \sqrt{4 \cdot 27 \cdot 5 \cdot 7} = 2 \cdot 3 \cdot \sqrt{3 \cdot 5 \cdot 7} = 6 \cdot \sqrt{105}
\][/tex]
Therefore, the simplified form of [tex]\( \sqrt{28} \cdot \sqrt{27} \cdot \sqrt{5} \)[/tex] is [tex]\( 6 \sqrt{105} \)[/tex].
Upon reviewing the provided options:
- [tex]\( 7 \sqrt{15} \)[/tex]
- [tex]\( 6 \sqrt{15} \)[/tex]
- [tex]\( 6 \sqrt{105} \)[/tex]
- [tex]\( 7 \sqrt{105} \)[/tex]
The correct answer is:
[tex]\( 6 \sqrt{105} \)[/tex]
