Sure, let's work through this step-by-step to simplify the expression:
Given:
[tex]\[ \sqrt[4]{4 y^3} \cdot \sqrt[4]{12 y^2} \][/tex]
### Step 1: Express each term in a simpler form
For the first term, [tex]\(\sqrt[4]{4 y^3}\)[/tex]:
[tex]\[ 4 y^3 = (2^2) y^3 \][/tex]
We can write this as:
[tex]\[ \sqrt[4]{4 y^3} = \sqrt[4]{2^2 y^3} = \sqrt[4]{2^2} \cdot \sqrt[4]{y^3} = (2)^{\frac{2}{4}} \cdot (y^3)^{\frac{1}{4}} \][/tex]
[tex]\[ (2)^{\frac{2}{4}} = 2^{\frac{1}{2}} = \sqrt{2} \][/tex]
[tex]\[ (y^3)^{\frac{1}{4}} = y^{\frac{3}{4}} \][/tex]
So,
[tex]\[ \sqrt[4]{4 y^3} = \sqrt{2} \cdot y^{\frac{3}{4}} \][/tex]
For the second term, [tex]\(\sqrt[4]{12 y^2}\)[/tex]:
[tex]\[ 12 y^2 = (2^2 \cdot 3) y^2 \][/tex]
We can write this as:
[tex]\[ \sqrt[4]{12 y^2} = \sqrt[4]{2^2 \cdot 3 \cdot y^2} = \sqrt[4]{2^2} \cdot \sqrt[4]{3} \cdot \sqrt[4]{y^2} \][/tex]
Again,
[tex]\[ \sqrt[4]{2^2} = (2)^{\frac{2}{4}} = \sqrt{2} \][/tex]
[tex]\[ \sqrt[4]{3} = 3^{\frac{1}{4}} \][/tex]
[tex]\[ \sqrt[4]{y^2} = (y^2)^{\frac{1}{4}} = y^{\frac{2}{4}} = y^{\frac{1}{2}} \][/tex]
So,
[tex]\[ \sqrt[4]{12 y^2} = \sqrt{2} \cdot 3^{\frac{1}{4}} \cdot y^{\frac{1}{2}} \][/tex]
### Step 2: Multiply the two expressions
Now, multiply these simplified forms together:
[tex]\[ \sqrt{2} \cdot y^{\frac{3}{4}} \cdot \sqrt{2} \cdot 3^{\frac{1}{4}} \cdot y^{\frac{1}{2}} \][/tex]
Combine the [tex]\(\sqrt{2}\)[/tex] terms:
[tex]\[ \sqrt{2} \cdot \sqrt{2} = 2 \][/tex]
Combine the [tex]\(y\)[/tex] terms:
[tex]\[ y^{\frac{3}{4}} \cdot y^{\frac{1}{2}} = y^{\frac{3}{4} + \frac{1}{2}} = y^{\frac{3}{4} + \frac{2}{4}} = y^{\frac{5}{4}} \][/tex]
Thus the final expression is:
[tex]\[ 2 \cdot 3^{\frac{1}{4}} \cdot y^{\frac{5}{4}} \][/tex]
Or equivalently:
[tex]\[ 2 \cdot 3^{\frac{1}{4}} \cdot \left(y^{2} y^{3}\right)^{\frac{1}{4}} \][/tex]
### Step 3: Present the Final Result
The simplified product of [tex]\(\sqrt[4]{4 y^3} \cdot \sqrt[4]{12 y^2}\)[/tex] is:
[tex]\[ 2 \cdot 3^{\frac{1}{4}} \cdot (y^2)^{\frac{1}{4}} \cdot (y^3)^{\frac{1}{4}}\][/tex]
So the given expression:
[tex]\[ \sqrt[4]{4 y^3} \cdot \sqrt[4]{12 y^2} = 2 \cdot 3^{\frac{1}{4}} \cdot (y^{2})^{\frac{1}{4}} \cdot (y^{3})^{\frac{1}{4}}. \][/tex]
