Squares, square roots and cubes for RRB Group D

Squares and cubes are not a big topic on their own, but they are everywhere. Simplification, mensuration, Pythagoras in geometry and number-system questions all assume you can write 23² or ∛2,744 without stopping. A candidate who knows the tables below saves a few seconds on many questions, and over a whole paper that adds up to extra time for the harder ones.

This guide gives you the tables to learn, the unit-digit rules that make roots quick, and the question types that come up directly.

Squares to learn

You probably know 1² to 10² already. Learn these next:

Cubes to learn

Unit-digit rules

The last digit of a square or cube tells you a lot about its root.

Quick squaring

Numbers ending in 5: multiply the first part by the next number, then write 25.

• 35²: 3 × 4 = 12, so 1,225.
• 65²: 6 × 7 = 42, so 4,225.
• 85²: 8 × 9 = 72, so 7,225.

Numbers near 50 or 100: use (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b².

• 52² = 2,500 + 200 + 4 = 2,704.
• 98² = 10,000 − 400 + 4 = 9,604.

Our BODMAS guide has more quick-multiplication identities.

Finding a square root quickly

1. Look at the unit digit to get two possible last digits.
2. Find which two tens the number lies between.
3. Decide between the two candidates. The number ending in 5 in that range is a useful middle point: if the number is above that square, take the larger candidate.

For numbers that are hard to place, use prime factors: pair up equal primes and take one from each pair. √1,764 = √(2² × 3² × 7²) = 2 × 3 × 7 = 42. See the number system guide for factorising.

Finding a cube root quickly

For a perfect cube of 4 to 6 digits:

1. The last three digits give the unit digit of the root (use the table).
2. The digits before those three tell you the tens digit: find the largest cube that is not more than that number.

Roots of decimals

Count decimal places. A square root halves the number of decimal places, so the number must have an even count.

• √0.0081 = √(81/10,000) = 9/100 = 0.09.
• √0.16 = 0.4.

Making a perfect square

Break the number into primes. For a perfect square, every prime must appear an even number of times.

• Smallest multiplier: multiply by the primes that are left unpaired.
• Smallest divisor: divide by those same unpaired primes.

Worked examples

Example 1: Find √2,304.

• Unit digit 4, so the root ends in 2 or 8.
• 40² = 1,600 and 50² = 2,500, so the root is between 40 and 50: 42 or 48.
• 45² = 2,025, and 2,304 is bigger, so take the larger: 48. Check: 48² = 2,304.

Example 2: Find ∛10,648.

• Last three digits 648 end in 8, so the root ends in 2.
• The part before is 10. The largest cube not more than 10 is 8 = 2³, so the tens digit is 2.
• Root = 22. Check: 22³ = 10,648.

Example 3: Find the smallest number by which 72 must be multiplied to make a perfect square.

• 72 = 2³ × 3². One 2 is unpaired.
• Multiply by 2: 144 = 12².

Example 4: Find the smallest number by which 3,888 must be divided to make a perfect square.

• 3,888 = 2⁴ × 3⁵. One 3 is unpaired.
• Divide by 3: 1,296 = 36².

Example 5: √0.16 + √0.0049

• √0.16 = 0.4 and √0.0049 = 0.07.
• 0.4 + 0.07 = 0.47.

Common mistakes

• Picking the wrong one of the two unit-digit candidates without checking against the "ends in 5" square.
• Getting the decimal places wrong in roots of decimals.
• Mixing up the cube root unit digits: a cube ending in 2 has a root ending in 8, and the reverse.
• Multiplying by the full number instead of only the unpaired prime.

Practice set

1. What is 45²?
2. Find √5,776.
3. Find ∛2,744.
4. Find √0.0144.
5. Find ∛29,791.
6. √169 + √144 − ∛125
7. What is the smallest number that must be subtracted from 1,000 to make a perfect square?
8. Which of these cannot be a perfect square: 1,156, 2,187, 3,025?

Answers:

1. 2,025. 4 × 5 = 20, then 25.
2. 76. Ends in 6, so 74 or 76; 75² = 5,625 is smaller than 5,776, so take 76.
3. 14. From the cube table.
4. 0.12. √(144/10,000) = 12/100.
5. 31. 791 ends in 1, so the unit digit is 1; 27 ≤ 29 < 64, so the tens digit is 3.
6. 20. 13 + 12 − 5.
7. 39. The largest square below 1,000 is 31² = 961; 1,000 − 961 = 39.
8. 2,187. It ends in 7, which no perfect square does. (1,156 = 34² and 3,025 = 55².)

What to do next

• Learn squares to 30 and cubes to 15, a few rows a day, until you can recall them without the table.
• Copy the unit-digit table onto a card and practise ten square roots and five cube roots with it.
• Solve five "smallest multiplier or divisor" questions with prime factors.
• Use these tables in BODMAS and simplification and in the maths plan.