>I followed the method shown by the textbook completely.

No you didn't. Notice what they did.

>√25 × √3 = 5√3

They had the square root of 25 and replaced it by 5. Not the square root of 5, just 5.

To "follow the method completely" you replace the square root of 16 by 4. Do you see how your statement:

>√16×√5 = √4×√5

has two radical signs while their example only has one after they do this?

>√25 × √3 = 5√3