Certainly! Let's simplify the given expression [tex]\( -9 a b^5 \left( a^2 b^4 - 10 a^3 b + 5 a b - 3 \right) \)[/tex] step by step.
1. Distribute the term outside the parenthesis:
The original expression is:
[tex]\[
-9 a b^5 \left( a^2 b^4 - 10 a^3 b + 5 a b - 3 \right)
\][/tex]
2. Break down the distribution over each term inside the parenthesis:
We will distribute [tex]\(-9 a b^5\)[/tex] to each term inside the parenthesis one by one:
1. First term: [tex]\(-9 a b^5 \cdot a^2 b^4\)[/tex]
2. Second term: [tex]\(-9 a b^5 \cdot (-10 a^3 b)\)[/tex]
3. Third term: [tex]\(-9 a b^5 \cdot 5 a b\)[/tex]
4. Fourth term: [tex]\(-9 a b^5 \cdot (-3)\)[/tex]
3. Multiply each term:
1. For the first term:
[tex]\[
-9 a b^5 \cdot a^2 b^4 = -9 a^{(1+2)} b^{(5+4)} = -9 a^3 b^9
\][/tex]
2. For the second term:
[tex]\[
-9 a b^5 \cdot (-10 a^3 b) = 90 a^{(1+3)} b^{(5+1)} = 90 a^4 b^6
\][/tex]
3. For the third term:
[tex]\[
-9 a b^5 \cdot 5 a b = -45 a^{(1+1)} b^{(5+1)} = -45 a^2 b^6
\][/tex]
4. For the fourth term:
[tex]\[
-9 a b^5 \cdot (-3) = 27 a b^5
\][/tex]
4. Combine the simplified terms:
By combining these terms, we get:
[tex]\[
-9 a^3 b^9 + 90 a^4 b^6 - 45 a^2 b^6 + 27 a b^5
\][/tex]
5. Factor out any common terms amongst the resulting expression:
We notice that each term can factor out [tex]\(9 a b^5\)[/tex]:
[tex]\[
-9 a^3 b^9 + 90 a^4 b^6 - 45 a^2 b^6 + 27 a b^5 = 9 a b^5 \left( 10 a^3 b - a^2 b^4 - 5 a b + 3 \right)
\][/tex]
Thus, the simplified version of the given expression is:
[tex]\[
9 a b^5 \left( 10 a^3 b - a^2 b^4 - 5 a b + 3 \right)
\][/tex]
This completes the detailed step-by-step simplification of the expression.
