Question:
If \[ x=3 \] find the value of:
\[ \left(\frac{5}{x} + 5x\right) \left(\frac{25}{x^2} – 25 + 25x^2\right) \]
Solution:
Rearranging the terms:
\[ \left(\frac{5}{x} + 5x\right) \left(\frac{25}{x^2} – \frac{25x}{x} + 25x^2\right) \]
Using identity:
\[ (a+b)(a^2-ab+b^2)=a^3+b^3 \]
Here, \[ a=\frac{5}{x},\qquad b=5x \]
\[ = \left(\frac{5}{x}\right)^3 + (5x)^3 \]
Substituting \[ x=3 \]
\[ = \left(\frac{5}{3}\right)^3 + (15)^3 \]
\[ = \frac{125}{27} + 3375 \]
\[ = \frac{125+91125}{27} \]
\[ = \frac{91250}{27} \]