To simplify the given expression, [tex]\(\frac{14 x^4 y^6}{7 x^4 y^2}\)[/tex], we can follow these steps:
1. Simplify the constants:
- Divide the coefficients: [tex]\(\frac{14}{7} = 2\)[/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
- We have [tex]\(x^4\)[/tex] in both the numerator and the denominator. When we divide [tex]\(x^4\)[/tex] by [tex]\(x^4\)[/tex], they cancel each other out: [tex]\(\frac{x^4}{x^4} = 1\)[/tex]
3. Simplify the [tex]\(y\)[/tex] terms:
- We have [tex]\(y^6\)[/tex] in the numerator and [tex]\(y^2\)[/tex] in the denominator. When we divide [tex]\(y^6\)[/tex] by [tex]\(y^2\)[/tex], we subtract the exponents: [tex]\(y^{6-2} = y^4\)[/tex]
Putting it all together, the simplified expression is:
[tex]\[ 2 \cdot y^4 \][/tex]
Now, let’s look at the given options and see which one matches our simplified expression:
- A. [tex]\(\frac{2 y^4}{x^4}\)[/tex] does not match because we do not have [tex]\(x^4\)[/tex] in the denominator
- B. [tex]\(\frac{7 y^3}{z^3}\)[/tex] does not match in any form
- C. [tex]\(7 x^4 y^4\)[/tex] does not match because the coefficients do not match (we have 2, not 7) and it still includes [tex]\(x^4\)[/tex]
- D. [tex]\(2 x^2 y^3\)[/tex] does not match because the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] do not match
The correct option that closely matches our simplified expression [tex]\(2 \cdot y^4\)[/tex] is:
[tex]\[ \boxed{2 \cdot y^4} \][/tex]
