To simplify [tex]\(\sqrt{6} \cdot \sqrt{30}\)[/tex], we can use a property of square roots. The property states that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. Let's apply this property step by step:
1. Identifying the individual square roots:
- [tex]\(\sqrt{6} \approx 2.449489742783178\)[/tex]
- [tex]\(\sqrt{30} \approx 5.477225575051661\)[/tex]
2. Using the property of square roots:
According to the property of square roots:
[tex]\[
\sqrt{6} \cdot \sqrt{30} = \sqrt{6 \cdot 30}
\][/tex]
3. Multiplying inside the square root:
Calculate the product inside the square root:
[tex]\[
6 \cdot 30 = 180
\][/tex]
4. Taking the square root of 180:
[tex]\[
\sqrt{180} \approx 13.416407864998739
\][/tex]
So, the simplified form of [tex]\(\sqrt{6} \cdot \sqrt{30}\)[/tex] is [tex]\(\sqrt{180} \approx 13.416407864998739\)[/tex].
To further check consistency with [tex]\(\sqrt{180}\)[/tex]:
- [tex]\(\sqrt{180} = \sqrt{6 \cdot 30}\)[/tex]
- Since [tex]\(\sqrt{6 \cdot 30} \approx 13.416407864998739\)[/tex]
Therefore, the simplified form of [tex]\(\sqrt{6} \cdot \sqrt{30}\)[/tex] is indeed [tex]\(\sqrt{180}\)[/tex], and numerically, it evaluates to approximately [tex]\(13.416407864998739\)[/tex].
