Certainly! Let's simplify the given expression step by step:
The expression we need to simplify is:
[tex]\[
\frac{1}{3} \sqrt{\frac{x}{y^2}} \cdot 6 \sqrt{\frac{2}{7}}
\][/tex]
### Step 1: Simplify Constants
First, let's simplify the constant terms:
[tex]\[
\frac{1}{3} \cdot 6 \cdot \sqrt{\frac{2}{7}}
\][/tex]
- Multiply [tex]\(\frac{1}{3}\)[/tex] by 6:
[tex]\[
\frac{1}{3} \cdot 6 = 2
\][/tex]
- Now, we have:
[tex]\[
2 \cdot \sqrt{\frac{2}{7}}
\][/tex]
### Step 2: Understand the Variable Part
Next, let's look at the variable part:
[tex]\[
\sqrt{\frac{x}{y^2}}
\][/tex]
This is already in its simplest form and cannot be simplified further algebraically.
### Step 3: Combine Constant and Variable Parts
Now, we'll combine the simplified constant and variable parts together:
[tex]\[
2 \sqrt{\frac{2}{7}} \cdot \sqrt{\frac{x}{y^2}}
\][/tex]
### Step 4: Use the Property of Square Roots
Recall the property of square roots: [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. Using this property, we can combine the square roots:
[tex]\[
2 \sqrt{\frac{2}{7} \cdot \frac{x}{y^2}} = 2 \sqrt{\frac{2x}{7y^2}}
\][/tex]
### Step 5: Simplify the Final Expression
Our final expression now combines the constant and variable parts:
[tex]\[
2 \sqrt{\frac{2x}{7y^2}}
\][/tex]
### Numerical Approximation
After performing these steps, we see that the constant multiplier before the variable part simplifies numerically to approximately [tex]\(1.0690449676496976\)[/tex]. Therefore, the final simplified expression is:
[tex]\[
1.0690449676496976 \cdot \sqrt{\frac{x}{y^2}}
\][/tex]
Or equivalently:
[tex]\[
1.0690449676497 \sqrt{\frac{x}{y^2}}
\][/tex]
Thus, the expression [tex]\(\frac{1}{3} \sqrt{\frac{x}{y^2}} \cdot 6 \sqrt{\frac{2}{7}}\)[/tex] simplifies to approximately [tex]\(1.0690449676497 \sqrt{\frac{x}{y^2}}\)[/tex].
