Sure, let's simplify the given expression step-by-step using algebraic rules. For each step, we'll make sure we clearly understand each part of the process.
First, let’s recall the expression we need to simplify:
[tex]\[
\frac{(4a^3 b^8 c^5 d^{-1} e f^{-4})^2 \cdot (19 a b c d^4)^0}{a^3 b^{-9} c d^{-1}}
\][/tex]
### Simplifying the Numerator
1. Simplify [tex]\((19 a b c d^4)^0\)[/tex]:
Anything raised to the power of zero is [tex]\(1\)[/tex]:
[tex]\[
(19 a b c d^4)^0 = 1
\][/tex]
So, the numerator simplifies to:
[tex]\[
(4a^3 b^8 c^5 d^{-1} e f^{-4})^2
\][/tex]
2. Expand the exponent in the numerator:
[tex]\[
(4a^3 b^8 c^5 d^{-1} e f^{-4})^2 = 4^2 (a^3)^2 (b^8)^2 (c^5)^2 (d^{-1})^2 e^2 (f^{-4})^2
\][/tex]
Simplified, this becomes:
[tex]\[
16 a^6 b^{16} c^{10} d^{-2} e^2 f^{-8}
\][/tex]
### Simplifying the Denominator
The denominator remains as:
[tex]\[
a^3 b^{-9} c d^{-1}
\][/tex]
### Combining the Numerator and Denominator
Now we combine the simplified numerator and denominator:
[tex]\[
\frac{16 a^6 b^{16} c^{10} d^{-2} e^2 f^{-8}}{a^3 b^{-9} c d^{-1}}
\][/tex]
### Simplifying the Expression
1. Combine the exponents for variable [tex]\(a\)[/tex]:
Using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
a^{6-3} = a^3
\][/tex]
2. Combine the exponents for variable [tex]\(b\)[/tex]:
[tex]\[
b^{16-(-9)} = b^{16+9} = b^{25}
\][/tex]
3. Combine the exponents for variable [tex]\(c\)[/tex]:
[tex]\[
c^{10-1} = c^9
\][/tex]
4. Combine the exponents for variable [tex]\(d\)[/tex]:
[tex]\[
d^{-2-(-1)} = d^{-2+1} = d^{-1}
\][/tex]
5. Combine [tex]\(e\)[/tex] and [tex]\(f\)[/tex]:
Variables [tex]\(e\)[/tex] and [tex]\(f\)[/tex] do not have equivalents in the denominator to simplify further:
[tex]\[
e^2 \quad \text{and} \quad f^{-8}
\][/tex]
So, the fully simplified expression is:
[tex]\[
16 a^3 b^{25} c^9 d^{-1} e^2 f^{-8}
\][/tex]
### Summary of Simplified Expression
The simplified form of the given expression:
[tex]\[
\boxed{16 a^3 b^{25} c^9 d^{-1} e^2 f^{-8}}
\][/tex]
This completes the step-by-step simplification.
