Squares and square roots for SSC GD

Squares and square roots rarely come as a whole chapter of questions, but they sit inside many others. A simplification sum with √1,764, a circle with area 616 cm², a right-angled triangle with sides 5 and 12: each is quick if you know your squares and slow if you don't.

The good news is that this is mostly memory plus two or three tricks. A few days of practice can save you time across the whole maths section.

Squares to memorise (1–30)

Cubes to memorise (1–15)

Quick ways to square a number

Numbers ending in 5. Multiply the first digit by the next number, then write 25 at the end. For 65²: 6 × 7 = 42, so 4,225.

Numbers near 50. Use (50 ± d)² = 2,500 ± 100d + d². For 48²: 2,500 − 200 + 4 = 2,304.

Numbers near 100. Take away the gap from the number, then write the gap squared as two digits. For 97²: gap 3, 97 − 3 = 94, 3² = 09, so 9,409.

Is it a perfect square?

• A perfect square never ends in 2, 3, 7 or 8.
• If it ends in zeros, the number of zeros is even (100, 3,600, 4,90,000).
• The digital root (repeated digit sum) of a perfect square is 1, 4, 7 or 9.

These rules let you strike out wrong options in seconds.

Finding square roots

Method 1: last digit plus range

Find which two tens the root lies between, use the last digit to get two choices, then decide using the square of the number ending in 5 in between.

Method 2: prime factors

Break the number into primes, pair them up and take one from each pair. This also works when you need to make a number a perfect square.

Cube roots of perfect cubes

For cubes, each last digit gives exactly one answer: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9. Then drop the last three digits and see which cube the rest lies above.

Decimals and fractions

• Count decimal places: the square has twice as many. √0.0049 = 0.07 because 0.07 has two places and 0.0049 has four.
• √1.44 = 1.2, √0.25 = 0.5, √0.09 = 0.3.
• For a fraction, take the root of top and bottom: √(169/225) = 13/15.

Solved examples

Example 1. Find 85².

1. 8 × 9 = 72.
2. Write 25 at the end: 7,225.

Example 2. Find √2,209.

1. 40² = 1,600 and 50² = 2,500, so the root is in the 40s.
2. It ends in 9, so the root ends in 3 or 7: 43 or 47.
3. 45² = 2,025, which is less than 2,209, so the root is above 45.
4. Answer: 47. Check: 47² = 2,209.

Example 3. Find √784 by factors.

1. 784 = 2 × 2 × 2 × 2 × 7 × 7.
2. Pairs: (2 × 2), (2 × 2), (7 × 7). One from each: 2 × 2 × 7.
3. Answer: 28.

Example 4. Find the cube root of 13,824.

1. Last digit 4, so the root ends in 4.
2. Drop the last three digits: 13. Since 2³ = 8 and 3³ = 27, the tens digit is 2.
3. Answer: 24. Check: 24³ = 13,824.

Example 5. Which of these is not a perfect square: 1,444, 2,025, 3,128, 4,096?

1. 3,128 ends in 8, and no perfect square ends in 8.
2. Answer: 3,128. (The others are 38², 45² and 64².)

Example 6. Find the smallest number by which 72 must be multiplied to get a perfect square.

1. 72 = 2 × 2 × 2 × 3 × 3.
2. The 3s pair up, and two of the 2s pair up. One 2 is left alone.
3. Multiply by 2. You get 144 = 12².

Practice set

1. Find 95².
2. Find 52².
3. Find √1,764.
4. Find √5,329.
5. Find √0.0144.
6. Find the cube root of 2,744.
7. Find √(256/361).
8. Find the smallest number by which 180 must be multiplied to get a perfect square.

Answers:

1. 9 × 10 = 90, then 25: 9,025.
2. 2,500 + 200 + 4 = 2,704.
3. Between 40 and 50, ends in 2 or 8; 45² = 2,025 is more, so 42.
4. Between 70 and 80, ends in 3 or 7; 75² = 5,625 is more, so 73.
5. √144 = 12, with two decimal places: 0.12.
6. Ends in 4. Dropping 744 leaves 2, which lies between 1³ and 2³, so the tens digit is 1: 14.
7. 16/19.
8. 180 = 2 × 2 × 3 × 3 × 5. The 5 is alone, so multiply by 5 to get 900 = 30².

What to do next

• Say the squares to 30 and cubes to 15 aloud every morning for ten days.
• Solve ten square-root questions using only the last-digit method.
• Use these numbers in simplification and mensuration practice.
• Practise the near-50 and near-100 tricks on ten numbers of your own.