Let's simplify the expression [tex]\(\frac{15 a^2 b^7}{25 a^7 b}\)[/tex] and identify the correct answer from the given options.
### Step-by-Step Simplification:
1. Write down the given expression:
[tex]\[
\frac{15 a^2 b^7}{25 a^7 b}
\][/tex]
2. Factorize the constants and the variables separately in the numerator and the denominator:
The constants in the numerator and denominator are [tex]\(15\)[/tex] and [tex]\(25\)[/tex] respectively. The variables include [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
3. Simplify the constants:
[tex]\[
\frac{15}{25} = \frac{3}{5}
\][/tex]
4. Simplify the variable [tex]\(a\)[/tex]:
In the numerator, we have [tex]\(a^2\)[/tex] and in the denominator, we have [tex]\(a^7\)[/tex].
[tex]\[
\frac{a^2}{a^7} = a^{2-7} = a^{-5} = \frac{1}{a^5}
\][/tex]
5. Simplify the variable [tex]\(b\)[/tex]:
In the numerator, we have [tex]\(b^7\)[/tex] and in the denominator, we have [tex]\(b\)[/tex].
[tex]\[
\frac{b^7}{b} = b^{7-1} = b^6
\][/tex]
6. Combine the simplified parts:
Putting it all together:
[tex]\[
\frac{3}{5} \cdot \frac{b^6}{a^5} = \frac{3b^6}{5a^5}
\][/tex]
### State the exclusions:
Since we are dealing with variables in the denominator, we need to make sure that the denominators are not zero to avoid undefined expressions. Therefore:
[tex]\[
a \neq 0 \quad \text{and} \quad b \neq 0
\][/tex]
### Conclusion:
The simplified expression is [tex]\(\frac{3b^6}{5a^5}\)[/tex] with the exclusions [tex]\(a \neq 0\)[/tex] and [tex]\(b \neq 0\)[/tex].
From the provided options, the correct answer is:
[tex]\[
\frac{3 b^6}{5 a^5} ; a \neq 0 \text{ and } b \neq 0
\][/tex]
