To divide the given expressions, we can rewrite the division as a multiplication by the reciprocal. The given expressions are:
[tex]\[
\frac{5 b}{4 a^2} \div \frac{b^4}{8 a^5 b}
\][/tex]
To handle this division, we'll follow these steps:
1. Rewrite the division as multiplication by the reciprocal of the second fraction.
[tex]\[
\frac{5 b}{4 a^2} \times \frac{8 a^5 b}{b^4}
\][/tex]
2. Multiply the numerators and the denominators.
[tex]\[
\frac{5 b \cdot 8 a^5 b}{4 a^2 \cdot b^4}
\][/tex]
3. Combine the numerators and the denominators.
[tex]\[
\frac{40 b^2 a^5}{4 a^2 b^4}
\][/tex]
4. Simplify the fraction by dividing the numerator and the denominator by their common factors.
- For the coefficients, divide 40 by 4:
[tex]\[
\frac{40}{4} = 10
\][/tex]
- For the [tex]\( b \)[/tex]-terms: [tex]\( b^2 \)[/tex] in the numerator and [tex]\( b^4 \)[/tex] in the denominator, subtract the exponents (since [tex]\( b^4 \)[/tex] in the denominator can be written as [tex]\( \frac{1}{b^4} \)[/tex] when the property of exponents [tex]\( \frac{b^m}{b^n} = b^{m-n} \)[/tex] is applied):
[tex]\[
b^{2-4} = b^{-2} = \frac{1}{b^2}
\][/tex]
- For the [tex]\( a \)[/tex]-terms: [tex]\( a^5 \)[/tex] in the numerator and [tex]\( a^2 \)[/tex] in the denominator, subtract the exponents:
[tex]\[
a^{5-2} = a^3
\][/tex]
5. Combine these simplified terms:
[tex]\[
10 \times \frac{a^3}{b^2}
\][/tex]
Thus, the simplified form of the division is:
[tex]\[
\frac{10 a^3}{b^2}
\][/tex]
This is the final simplified result.
