To solve the expression [tex]\(\sqrt{12} \cdot \sqrt{y^3} \cdot \sqrt{6y}\)[/tex] and simplify it by removing all perfect squares from inside the square roots, let's go through the steps in detail:
1. Combine the Square Roots:
Start by combining the square roots into a single square root.
[tex]\[
\sqrt{12} \cdot \sqrt{y^3} \cdot \sqrt{6y} = \sqrt{12 \cdot y^3 \cdot 6y}
\][/tex]
2. Simplify the Expression Inside the Square Root:
Multiply the terms inside the square root.
\begin{align}
12 \cdot y^3 \cdot 6y &= (12 \cdot 6) \cdot (y^3 \cdot y) \\
&= 72 \cdot y^4
\end{align}
So, the expression now is:
[tex]\[
\sqrt{12 \cdot y^3 \cdot 6y} = \sqrt{72y^4}
\][/tex]
3. Remove Perfect Squares from Inside the Square Root:
Factorize [tex]\(72\)[/tex] to identify and remove perfect squares:
[tex]\[
72 = 36 \times 2
\][/tex]
Notice that [tex]\(36\)[/tex] is a perfect square:
[tex]\[
72y^4 = 36 \times 2 \times y^4
\][/tex]
Thus,
[tex]\[
\sqrt{72y^4} = \sqrt{36 \times 2 \times y^4}
\][/tex]
Separate the perfect squares inside the square root:
[tex]\[
\sqrt{36 \times 2 \times y^4} = \sqrt{36} \times \sqrt{2} \times \sqrt{y^4}
\][/tex]
4. Simplify the Square Roots:
Evaluate the square roots of the perfect squares:
[tex]\[
\sqrt{36} = 6 \quad \text{and} \quad \sqrt{y^4} = y^2
\][/tex]
Thus,
[tex]\[
\sqrt{36} \times \sqrt{2} \times \sqrt{y^4} = 6 \times \sqrt{2} \times y^2
\][/tex]
5. Combine the Results:
Finally, combine the terms to get the simplified form:
[tex]\[
6 \times \sqrt{2} \times y^2
\][/tex]
So, the expression simplifies to:
[tex]\[
\boxed{6\sqrt{2}y^2}
\][/tex]
