Squares, Square Roots, Cubes and Cube Roots
Squares, Square Roots, Cubes and Cube Roots
This topic gives 2 to 3 questions and is the most pattern-driven part of the number system. Almost every question can be solved by recognising a property rather than calculating, which is why it rewards preparation more than speed.
Properties of Perfect Squares
A perfect square is a number obtained by multiplying an integer by itself. Several properties let you reject options instantly.
- A perfect square never ends in 2, 3, 7 or 8.
- A perfect square has an even number of zeros at the end, or none.
- The square of an even number is even; of an odd number, odd.
- The number of digits in the square root of an n-digit number is n/2 rounded up.
- The sum of the first n odd numbers is n²: 1 + 3 + 5 + 7 = 16.
That last property is worth teaching as well as knowing, because it shows a class 8 student why squares grow the way they do.
✗ 1,458 could be a perfect square | ✓ It ends in 8, so it cannot be a perfect square
Useful Identities
(a + b)² = a² + 2ab + b² | (a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
These turn hard arithmetic into easy arithmetic. To find 103², use (100 + 3)² = 10000 + 600 + 9 = 10609. To find 51² − 49², use (51 + 49)(51 − 49) = 100 × 2 = 200, with no squaring at all.
Finding Square Roots
Two methods are prescribed and both can be asked by name.
- Prime factorisation — pair the primes and take one from each pair. For 1,296 = 2⁴ × 3⁴, the root is 2² × 3² = 36.
- Long division method — used when the number is large or not a perfect square.
If a number is not a perfect square, its square root is irrational, which links this topic straight back to the rational and irrational topic earlier in the chapter.
Cubes and Cube Roots
| n | n² | n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1000 |
Unlike squares, a cube keeps the sign of the number: the cube of a negative is negative, so cube roots of negative numbers exist among the reals while square roots do not. Learning the cubes to 10 makes cube-root questions instant, because the unit digit of the cube identifies the unit digit of the root.
वर्ग कभी 2, 3, 7 या 8 पर समाप्त नहीं होता — इससे विकल्प तुरंत कटते हैं।
TRE pointer: The unit-digit rule is the fastest elimination tool in the entire maths paper: any option ending in 2, 3, 7 or 8 cannot be a perfect square, which often removes two of the five choices before you calculate anything. The sum of the first n odd numbers equals n² is the property most often asked in a teaching paper, because it is the visual proof used in the class 8 textbook. Elimination is also the honest way to raise your odds when a wrong answer costs −1/3.
60-Second Recap
- Perfect squares never end in 2, 3, 7 or 8, and end in an even number of zeros.
- Sum of the first n odd numbers equals n squared.
- (a+b)², (a−b)² and a²−b² turn hard arithmetic into easy arithmetic.
- Square roots by prime factorisation for perfect squares, long division otherwise.
- The square root of a non-perfect square is irrational.
- Cubes keep the sign, so negative numbers have real cube roots but not square roots.
Frequently Asked Questions
How can I tell quickly that a number is not a perfect square?
Check its last digit. A perfect square can never end in two, three, seven or eight. Also, a perfect square ends in an even number of zeros, so a number ending in a single zero is out. These two checks eliminate most wrong options in seconds.
What is the sum of the first n odd numbers?
It equals n squared. One plus three is four, plus five is nine, plus seven is sixteen. This pattern is used in the class 8 textbook as a visual proof of how squares grow, and teaching papers ask it because it tests understanding rather than calculation.
Why do negative numbers have cube roots but not square roots?
Because multiplying three negatives keeps the result negative, so the cube of negative two is negative eight and its cube root is negative two. Squaring, by contrast, always gives a positive result, so no real number squares to a negative value.
Which method should I use to find a square root in the exam?
Use prime factorisation when the number is clearly a perfect square and factorises easily, since pairing primes is quick and less error-prone. Use the long division method for large numbers or when the number is not a perfect square and an approximate value is required.