To simplify the expression [tex]\(\sqrt{12} \cdot \sqrt{y^3} \cdot \sqrt{6 y}\)[/tex], we follow these steps:
### Step 1: Combine the Square Roots
First, we use the property of square roots that states [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)[/tex]. Applying this property, we combine the square roots into a single square root:
[tex]\[
\sqrt{12} \cdot \sqrt{y^3} \cdot \sqrt{6 y} = \sqrt{12 \cdot y^3 \cdot 6y}
\][/tex]
### Step 2: Simplify Inside the Square Root
Next, we simplify the expression inside the square root. We proceed by multiplying the terms:
[tex]\[
12 \cdot 6 \cdot y^3 \cdot y = 72y^4
\][/tex]
Thus,
[tex]\[
\sqrt{12 \cdot y^3 \cdot 6 y} = \sqrt{72y^4}
\][/tex]
### Step 3: Factorize Inside the Square Root
We now factorize [tex]\(72y^4\)[/tex] to identify the perfect squares inside the square root. We know [tex]\(72 = 36 \cdot 2 = 6^2 \cdot 2\)[/tex], and [tex]\(y^4\)[/tex] is already a perfect square:
[tex]\[
72y^4 = (6^2 \cdot 2) \cdot y^4 = 36 \cdot 2 \cdot y^4
\][/tex]
### Step 4: Simplify the Square Root
We use the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Here, [tex]\(36 \cdot y^4\)[/tex] are perfect squares:
[tex]\[
\sqrt{72y^4} = \sqrt{36 \cdot 2 \cdot y^4} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{y^4}
\][/tex]
Since [tex]\(\sqrt{36} = 6\)[/tex] and [tex]\(\sqrt{y^4} = y^2\)[/tex], we get:
[tex]\[
\sqrt{36} \cdot \sqrt{2} \cdot \sqrt{y^4} = 6 \cdot \sqrt{2} \cdot y^2
\][/tex]
### Conclusion
Thus, the simplified form of the given expression is:
[tex]\[
\sqrt{12} \cdot \sqrt{y^3} \cdot \sqrt{6 y} = 6\sqrt{2}y^2
\][/tex]
So, the final answer is:
[tex]\[
6\sqrt{2}y^2
\][/tex]
