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Why This Chapter Is Pure Free Marks
Squares, cubes and roots are not a standalone topic in most exams — they hide inside percentage, mensuration, algebra and Data Interpretation questions, where a two-digit square or a perfect-cube identification is one step buried in a longer calculation. A student who has to multiply 47 × 47 by hand loses forty seconds that a student who recognises the base-method shortcut spends on the next question instead.
The good news is that this entire chapter runs on four or five repeatable tricks, not memorisation of hundreds of values. Learn the base method for squaring, the 'ends in 5' trick, the last-digit rule for identifying roots, and the algebraic expansion for cubes, and almost every question in this chapter collapses to a fifteen-second mental calculation — which is exactly why exams keep asking it.
1. Using the base method, what is 47²?
2. Treating 96 as (100 − 4), what is 96²?
3. Using the ends-in-5 shortcut, what is 65²?
4. Splitting 32 as (30 + 2), what is 32³?
5. 3375 is a perfect cube. Using the unit-digit map and digit-count bracket, what is its cube root?
6. 5929 is a perfect square. Using the digit-count and last-digit method, what is its square root?
7. Which of these can never be the last digit of a perfect square?
8. Using the base method with 1000 as the base, what is 998²?