Square root and cube root questions are among the fastest scoring items in AFCAT Numerical Ability, yet candidates waste minutes on long division. This page from The Cavalier teaches you the prime-factorisation method, the division method, and the powerful unit-digit shortcut so you can read off answers for perfect squares and cubes almost instantly and move on.
What square root and cube root mean
The square root of a number is the value which, when multiplied by itself, gives that number. If x2 = N, then x = √N. For example, since 92 = 81, √81 = 9. The symbol √ is the radical sign.
The cube root of a number is the value which, when multiplied by itself three times, gives that number. If x3 = N, then x = 3√N. Since 43 = 64, the cube root of 64 is 4.
A square root has two values, positive and negative (√81 = ±9), but in AFCAT the principal (positive) root is taken unless stated otherwise. A cube root keeps the sign of the original number, so the cube root of −27 is −3.
A perfect square is a number whose square root is a whole number (1, 4, 9, 16, 25, 36...). A perfect cube is a number whose cube root is a whole number (1, 8, 27, 64, 125...). Recognising these on sight is half the battle in the exam.
It also helps to know how roots behave. Squaring always produces a non-negative result, which is why no real square root exists for a negative number; AFCAT therefore keeps square-root questions to positive numbers. Cubing, by contrast, preserves sign, so cube roots are defined for every real number including negatives. Index notation ties these ideas together: √N is N raised to the power one-half, and 3√N is N raised to the power one-third. Seeing roots as fractional powers makes laws such as √N × √N = N feel natural rather than memorised.
Tables you must memorise
Speed in this topic comes from memory, not calculation. Before exam day, lock in these values so you recognise them instantly.
- Squares 1 to 30: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.
- Cubes 1 to 15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375.
If you know squares up to 30 and cubes up to 15, most AFCAT root questions reduce to instant recall. The numbers chosen in the paper almost always sit inside these ranges.
A perfect square never ends in 2, 3, 7 or 8. So 4358 or 2087 cannot be perfect squares, no matter how they look. This single rule lets you eliminate wrong options at a glance.
Square root by prime factorisation
This is the most reliable method for perfect squares. Break the number into prime factors, pair identical factors, and take one from each pair.
Find √1764.
Students take the product of all prime factors instead of one factor per pair. After pairing, multiply ONE element from each pair only. Forgetting this turns √1764 into 1764, a guaranteed wrong answer.
Prime factorisation also tells you instantly whether a number is a perfect square at all: every prime must appear an even number of times. If one prime is left unpaired, the number is not a perfect square, and the smallest factor you must multiply or divide by to make it one is exactly that unpaired prime. For instance 1080 = 23 × 33 × 5 has unpaired 2, 3 and 5, so the smallest multiplier making it a perfect square is 2 × 3 × 5 = 30. AFCAT frequently asks for this least multiplier or divisor, so practise reading the factor exponents directly.
Square root by the division method
Use this when the number is large or not an obvious perfect square. It also gives roots correct to decimals.
- Pair the digits from the right (for the integer part). Place a bar over each pair.
- Find the largest number whose square is ≤ the leftmost pair. This is the first digit of the root.
- Subtract, bring down the next pair, and double the current quotient to form the new divisor's tens part.
- Choose a units digit so that (divisor with that digit) × (that digit) is ≤ the current number.
- Repeat until pairs are exhausted.
Find √5329 by division.
For decimals, pair digits both sides of the decimal point: √5.29 pairs as 5 . 29 and gives 2.3. Always pair away from the decimal point, not from the left edge.
The unit-digit shortcut for square roots
For a perfect square with up to 4 digits, you can find the root in under ten seconds without any division.
- Look at the last digit to fix the unit digit of the root: ends in 1→1 or 9; 4→2 or 8; 5→5; 6→4 or 6; 9→3 or 7; 0→0.
- Strip the last two digits and look at the remaining left part. Find the largest square ≤ it; its root is the tens digit.
- Decide between the two options using the nearest tens-square boundary.
Find √3249 mentally.
The decider step uses the midpoint square (here 552). If N is above the midpoint square, take the larger candidate; if below, take the smaller. This resolves the 1-or-9, 3-or-7, 2-or-8, 4-or-6 ambiguity every time.
Cube root by prime factorisation
The same logic as square roots, but group factors in threes instead of pairs.
Find 3√3375.
Grouping cube-root factors in pairs instead of triples. A cube root needs three identical factors per group; two is wrong and gives an inflated answer.
Just as with squares, the exponents reveal whether a number is a perfect cube: every prime must appear a number of times that is a multiple of three. If a prime appears twice, you need one more of it to complete the triple. So to make 1715 = 5 × 73 a perfect cube you multiply by 52 = 25, because the single 5 needs two more copies. This least-multiplier idea is identical to the square case but counts to three instead of two, and AFCAT mixes both versions in the same paper to test whether you are paying attention.
The lightning cube-root trick
For perfect cubes up to 6 digits, this method beats every other approach. It works because each digit 0–9 gives a unique cube unit digit.
- Unit-digit map: cube ends in 0→0, 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9.
- Notice 2↔8 and 3↔7 swap; 0,1,4,5,6,9 map to themselves.
- Split off the last three digits. The last digit of the cube fixes the unit digit of the root using the map above.
- Take the remaining left group. Find the largest cube ≤ it; its root is the tens digit.
Find 3√110592 mentally.
No midpoint check is needed for cubes because every unit digit is unique. Once you read the last digit and the left group, the answer is fixed. This is the single fastest trick in the chapter.
Roots of decimals, fractions and surds
AFCAT mixes in decimals and surds, so know these rules cold.
- Fractions: √(a/b) = √a ÷ √b. So √(16/25) = 4/5.
- Products: √(a × b) = √a × √b. Use this to simplify √72 = √(36 × 2) = 6√2.
- Decimals: count decimal places. A perfect-square decimal has an even number of decimal places; its root has half as many. √0.0064 = 0.08.
Writing √0.0064 = 0.8 by halving only the visible digits. Count places carefully: 0.0064 has 4 decimal places, so the root has 2, giving 0.08, not 0.8.
To rationalise a surd denominator, multiply top and bottom by the surd: 1 ÷ √2 = √2 ÷ 2 ≈ 0.707. Memorise √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 for fast approximation questions.
When a question stacks several radicals, simplify from the inside out and pull perfect-square factors outside the sign one step at a time. For a nested expression such as √(2 × √(16)), first resolve the inner root, √16 = 4, then evaluate √(2 × 4) = √8 = 2√2. Treat surds like 2√3 as ordinary numbers you can add only when the part under the root matches: 2√3 + 5√3 = 7√3, but 2√3 + 5√2 cannot be combined. Keeping like and unlike surds straight prevents the most common slip in this part of the paper.
Estimating non-perfect roots
When a number is not a perfect square, AFCAT usually wants the nearest integer or a value between two whole numbers. Bracket it between known squares.
Estimate √150.
The gap between consecutive squares n2 and (n+1)2 is 2n+1. Use the fraction (N − n2) ÷ (2n+1) added to n for a quick first-decimal estimate.
The same bracketing idea works for cube roots: enclose the number between two consecutive cubes and judge how close it sits. To estimate 3√200, note 53 = 125 and 63 = 216; since 200 is near 216, the cube root is roughly 5.85. AFCAT options are usually spaced far enough apart that a one-decimal estimate is enough to pick the right answer, so resist the urge to compute beyond the precision the options demand. Time saved here can be spent on heavier algebra or data-interpretation questions elsewhere in the section.
Exam-style practice and recap
Q. The value of 3√1.728 is:
Answer: 1.728 = 1728/1000. We know 3√1728 = 12 (since 123 = 1728) and 3√1000 = 10. So 3√1.728 = 12/10 = 1.2.
For decimal cube roots, count decimal places in groups of three. 1.728 has 3 decimal places, so its cube root has 1 decimal place, confirming 1.2 over 12.
- Perfect squares never end in 2, 3, 7 or 8; use this to eliminate options.
- Square root by factorisation: pair factors, take one per pair. Cube root: group in threes.
- Unit-digit trick: last digit fixes the root's unit digit; left group gives the tens digit.
- For squares, resolve the two candidates using the midpoint (tens.5)2; cubes need no check.
- Decimals: even places for squares, multiples of three for cubes; root has half / one-third the places.
- Memorise squares to 30 and cubes to 15 for instant recall.
Frequently asked questions
How many square root and cube root questions appear in AFCAT?
Typically one to three questions across Numerical Ability, often blended into simplification or approximation items. They are quick marks if you have memorised squares to 30 and cubes to 15 and can apply the unit-digit shortcut.
Is the unit-digit trick reliable for all numbers?
It is fully reliable for perfect squares (up to 4 digits) and perfect cubes (up to 6 digits), which is what AFCAT uses. For non-perfect numbers, switch to the bracketing or division method to estimate the answer.
How do I quickly tell if a large number is a perfect square?
First check the last digit: if it ends in 2, 3, 7 or 8 it is never a perfect square. Also, a perfect square's digital root (repeated digit sum) is only 1, 4, 7 or 9. These two filters reject most non-squares instantly.
What is the difference in grouping for square roots versus cube roots?
In prime factorisation, square roots group identical factors in pairs and you take one factor per pair. Cube roots group factors in threes and take one factor per triple. Mixing these up is the most common error.
How should I handle decimal places under a root?
For square roots the number must have an even count of decimal places, and the root gets half that count. For cube roots the count must be a multiple of three, and the root gets one-third. Count carefully, for example the square root of 0.0064 is 0.08, not 0.8.
Related AFCAT Numerical Ability topics
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