Sure, let's simplify the given expression step-by-step:
The expression is:
[tex]\[
\frac{2 x^{\sqrt{2}} y^4 \cdot 4 x^2 y^4 \cdot 3 x}{3 x^{-3} y^2}
\][/tex]
### Step 1: Combine the Numerator Terms
First, we will combine all the terms in the numerator:
[tex]\[
2 x^{\sqrt{2}} y^4 \cdot 4 x^2 y^4 \cdot 3 x
\][/tex]
- Combine the constants: [tex]\( 2 \cdot 4 \cdot 3 = 24 \)[/tex]
- Combine the [tex]\( x \)[/tex]-terms: [tex]\( x^{\sqrt{2}} \cdot x^2 \cdot x = x^{\sqrt{2} + 2 + 1} = x^{\sqrt{2} + 3} \)[/tex]
- Combine the [tex]\( y \)[/tex]-terms: [tex]\( y^4 \cdot y^4 = y^{8} \)[/tex]
Thus, the numerator simplifies to:
[tex]\[
24 x^{\sqrt{2} + 3} y^8
\][/tex]
### Step 2: Simplify the Denominator
Now, let's look at the denominator:
[tex]\[
3 x^{-3} y^2
\][/tex]
### Step 3: Divide the Numerator by the Denominator
Next, we will divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{24 x^{\sqrt{2} + 3} y^8}{3 x^{-3} y^2}
\][/tex]
- Divide the constants: [tex]\( \frac{24}{3} = 8 \)[/tex]
- Divide the [tex]\( x \)[/tex]-terms: [tex]\( \frac{x^{\sqrt{2} + 3}}{x^{-3}} = x^{(\sqrt{2} + 3 + 3)} = x^{\sqrt{2} + 6} \)[/tex]
- Divide the [tex]\( y \)[/tex]-terms: [tex]\( \frac{y^8}{y^2} = y^{8 - 2} = y^6 \)[/tex]
Thus, the final expression simplifies to:
[tex]\[
8 x^{\sqrt{2} + 6} y^6
\][/tex]
### Step 4: Numerical Evaluation of the Exponent [tex]\(\sqrt{2} + 6\)[/tex]
We know that [tex]\(\sqrt{2} \approx 1.41421356237309\)[/tex]. Adding 6 to this value, we get:
[tex]\[
\sqrt{2} + 6 \approx 1.41421356237309 + 6 \approx 7.41421356237309
\][/tex]
### Final Expression
Therefore, the simplified expression is:
[tex]\[
8 x^{7.41421356237309} y^6
\][/tex]
So, the final result after simplification is:
[tex]\[
8 x^{7.41421356237309} y^6
\][/tex]
