Sure, let's simplify the given expression step by step:
Given expression:
[tex]\[
\frac{(4a)^3 \cdot a \cdot b + (2a)^5}{8a^2b + 8a^2b}
\][/tex]
Step 1: Simplify the numerator
1. [tex]\((4a)^3 \cdot a \cdot b\)[/tex]:
[tex]\[
(4a)^3 = 4^3 \cdot a^3 = 64a^3
\][/tex]
Therefore:
[tex]\[
(4a)^3 \cdot a \cdot b = 64a^3 \cdot a \cdot b = 64a^4b
\][/tex]
2. [tex]\((2a)^5\)[/tex]:
[tex]\[
(2a)^5 = 2^5 \cdot a^5 = 32a^5
\][/tex]
Combining these results:
[tex]\[
(4a)^3 \cdot a \cdot b + (2a)^5 = 64a^4b + 32a^5
\][/tex]
Step 2: Simplify the denominator
[tex]\[
8a^2b + 8a^2b = 16a^2b
\][/tex]
Step 3: Combine the simplified numerator and denominator
[tex]\[
\frac{64a^4b + 32a^5}{16a^2b}
\][/tex]
Step 4: Factor and simplify the expression
The numerator [tex]\(64a^4b + 32a^5\)[/tex] has a common factor. We can factor out [tex]\(32a^4\)[/tex]:
[tex]\[
64a^4b + 32a^5 = 32a^4(2b + a)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{32a^4(2b + a)}{16a^2b}
\][/tex]
Step 5: Cancel common factors in the numerator and denominator
[tex]\[
\frac{32a^4(2b + a)}{16a^2b} = \frac{32}{16} \cdot \frac{a^4}{a^2} \cdot \frac{(2b + a)}{b}
\][/tex]
Simplifying each part:
[tex]\[
\frac{32}{16} = 2
\][/tex]
[tex]\[
\frac{a^4}{a^2} = a^2
\][/tex]
This yields:
[tex]\[
\frac{2a^2(2b + a)}{b}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{2a^2 (2b + a)}{b}
\][/tex]
This is the final simplified form of the given expression.
