To simplify the expression [tex]\(\frac{15 a^6 b c^4}{35 a^2 c^4}\)[/tex], we can follow these steps:
1. Simplify the Coefficients:
The numerical part of the expression is [tex]\(\frac{15}{35}\)[/tex]. Simplifying this fraction:
[tex]\[
\frac{15}{35} = \frac{3 \cdot 5}{7 \cdot 5} = \frac{3}{7}
\][/tex]
2. Simplify the Powers of Variables:
- For [tex]\(a\)[/tex]:
Since the terms have [tex]\(a^6\)[/tex] in the numerator and [tex]\(a^2\)[/tex] in the denominator, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
a^{6-2} = a^4
\][/tex]
- For [tex]\(c\)[/tex]:
Since we have [tex]\(c^4\)[/tex] in both the numerator and the denominator, the exponents cancel each other out:
[tex]\[
c^{4-4} = c^0 = 1
\][/tex]
Therefore, [tex]\(c\)[/tex] is eliminated from the expression.
- For [tex]\(b\)[/tex]:
The variable [tex]\(b\)[/tex] is present only in the numerator, so it stays as it is.
3. Combine the Simplified Components:
Now, we combine the simplified parts:
[tex]\[
\frac{15 a^6 b c^4}{35 a^2 c^4} = \frac{3}{7} a^4 b
\][/tex]
Hence, the simplified expression is:
[tex]\[
\frac{3 a^4 b}{7}
\][/tex]
The correct choice from the given options is [tex]\(\frac{3 a^4 b}{7}\)[/tex].
