Certainly! Let's simplify [tex]\( \sqrt{6} \cdot \sqrt{30} \)[/tex] step-by-step.
1. Step 1: Multiply the radicands together
The product of two square roots can be expressed as the square root of the product of the radicands (the numbers inside the square roots). Therefore:
[tex]\[
\sqrt{6} \cdot \sqrt{30} = \sqrt{6 \times 30}
\][/tex]
2. Step 2: Calculate the product inside the square root
Multiply the numbers inside the square root:
[tex]\[
6 \times 30 = 180
\][/tex]
3. Step 3: Simplify the square root
Now, we have:
[tex]\[
\sqrt{6 \cdot 30} = \sqrt{180}
\][/tex]
Simplify [tex]\(\sqrt{180}\)[/tex]:
First, we can factorize 180 into its prime factors:
[tex]\[
180 = 2^2 \times 3^2 \times 5
\][/tex]
Using the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{5} = 2 \cdot 3 \cdot \sqrt{5} = 6 \sqrt{5}
\][/tex]
However, the exact simplified form in decimal value of [tex]\( \sqrt{180} \)[/tex] is:
[tex]\[
\sqrt{180} \approx 13.416407864998739
\][/tex]
Therefore, the steps verify that:
[tex]\[
\sqrt{6} \cdot \sqrt{30} = \sqrt{180} \approx 13.416407864998739
\][/tex]
So the correct and simplified form of [tex]\( \sqrt{6} \cdot \sqrt{30} \)[/tex] is [tex]\( \approx 13.416407864998739 \)[/tex].
