Simplify the Following Products
\[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) \left(\frac{1}{4}a^2 + 9b^2\right) \]
Solution:
Using identity:
\[ (a-b)(a+b)=a^2-b^2 \]
\[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) = \left(\frac{1}{2}a\right)^2 – (3b)^2 \]
\[ = \frac{1}{4}a^2 – 9b^2 \]
Now the expression becomes:
\[ \left(\frac{1}{4}a^2 – 9b^2\right) \left(\frac{1}{4}a^2 + 9b^2\right) \]
Again using identity:
\[ (a-b)(a+b)=a^2-b^2 \]
\[ = \left(\frac{1}{4}a^2\right)^2 – (9b^2)^2 \]
\[ = \frac{1}{16}a^4 – 81b^4 \]