Question:
If \[ y=1 \] find the value of:
\[ \left(5y + \frac{15}{y}\right) \left(25y^2 – 75 + \frac{225}{y^2}\right) \]
Solution:
Rearranging the terms:
\[ \left(5y + \frac{15}{y}\right) \left(25y^2 – 5y\cdot\frac{15}{y} + \frac{225}{y^2}\right) \]
Using identity:
\[ (a+b)(a^2-ab+b^2)=a^3+b^3 \]
Here, \[ a=5y,\qquad b=\frac{15}{y} \]
\[ = (5y)^3 + \left(\frac{15}{y}\right)^3 \]
Substituting \[ y=1 \]
\[ = (5)^3 + (15)^3 \]
\[ = 125+3375 \]
\[ =3500 \]