To determine which expression is equivalent to the given expression, we start with the given expression:
[tex]\[
\frac{12 x^8 y^4}{6 (x + 1)}
\][/tex]
First, let's simplify this expression step-by-step.
1. Simplify the coefficients:
The coefficient in the numerator is 12, and in the denominator, it is 6. We can simplify the ratio of these coefficients:
[tex]\[
\frac{12}{6} = 2
\][/tex]
2. Write the simplified expression:
After simplifying the coefficient, we can write:
[tex]\[
\frac{2 x^8 y^4}{x + 1}
\][/tex]
Next, let's check each provided option to see if it matches this simplified form:
- Option A: [tex]\(2 x^3 y^2\)[/tex]
This expression is clearly different from [tex]\(\frac{2 x^8 y^4}{x + 1}\)[/tex] because it does not have a denominator and the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] do not match.
- Option B: [tex]\(\frac{2}{x^2 y}\)[/tex]
This again does not match the form of [tex]\(\frac{2 x^8 y^4}{x + 1}\)[/tex]. The numerator is a constant (2) and not [tex]\((2 x^8 y^4)\)[/tex], and the denominator contains different variables and powers.
- Option C: [tex]\(\frac{2}{x^2 + 1}\)[/tex]
This matches the constant [tex]\(2\)[/tex] in the numerator but has a different denominator configuration compared to [tex]\(\frac{2 x^8 y^4}{x + 1}\)[/tex]. Hence, this option is not equivalent.
- Option D: [tex]\(2 x^6 y^2\)[/tex]
This expression is also different as it does not have a denominator and the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are not what we have in the simplified form.
To verify the correct answer rigorously, let's substitute specific values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] and compute the original expression and the values of each option:
Let's choose [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex]:
[tex]\[
\text{Original Expression:} \quad \frac{12 \cdot 2^8 \cdot 3^4}{6 \cdot (2 + 1)}
\][/tex]
[tex]\[
\Rightarrow \frac{12 \cdot 256 \cdot 81}{6 \cdot 3} = \frac{248832}{18} = 13824
\][/tex]
Now, let's compute the values for each given option:
- Option A: [tex]\(2 \cdot 2^3 \cdot 3^2 = 2 \cdot 8 \cdot 9 = 144\)[/tex]
- Option B: [tex]\(\frac{2}{2^2 \cdot 3} = \frac{2}{4 \cdot 3} = \frac{2}{12} = 0.1667\)[/tex] (approximately)
- Option C: [tex]\(\frac{2}{2^2 + 1} = \frac{2}{4 + 1} = \frac{2}{5} = 0.4\)[/tex]
- Option D: [tex]\(2 \cdot 2^6 \cdot 3^2 = 2 \cdot 64 \cdot 9 = 1152\)[/tex]
Comparing all these results with the computed original expression value [tex]\(13824\)[/tex], none of the options match exactly.
Therefore, the correct conclusion is that none of the provided options (A, B, C, or D) are equivalent to the given expression
[tex]\[
\frac{12 x^8 y^4}{6 (x + 1)}
\][/tex]
