Square Root Examples for Beginners

The best way to get comfortable with square roots is to work through plenty of examples, moving from the easy perfect squares to trickier decimals. This beginner-friendly reference gives you graded, fully worked examples in the NCERT Class 8 style, so you can see the patterns and build confidence for school tests and competitive exams alike. Keep a pen handy and try each one yourself before reading the solution.

Level 1: Easy Perfect Squares

These are the roots every student should know instantly. Notice how each answer, multiplied by itself, gives the original number.

• √4 = 2, because 2 × 2 = 4
• √9 = 3, because 3 × 3 = 9
• √25 = 5, because 5 × 5 = 25
• √64 = 8, because 8 × 8 = 64
• √100 = 10, because 10 × 10 = 100

Once these feel automatic, you have the anchors you need to estimate any other root. If you want to check a batch quickly, a square root calculator confirms them instantly.

Level 2: Larger Perfect Squares with Factorisation

Example: √324

Break 324 into primes: 324 = 2 × 2 × 3 × 3 × 3 × 3. Pair them: (2×2)(3×3)(3×3). Take one from each pair: 2 × 3 × 3 = 18. So √324 = 18.

Example: √729

729 = 3 × 3 × 3 × 3 × 3 × 3 = (3×3)(3×3)(3×3). One from each pair: 3 × 3 × 3 = 27. So √729 = 27. The factoring skill here is the same one a HCF and GCD calculator uses to break numbers apart.

Key takeaway: For any perfect square, the answer is just the product of one prime from each matching pair. If a prime has no pair, the number is not a perfect square and you switch to long division.

Level 3: Non-Perfect Squares

These have decimal answers. You can estimate them between perfect squares or compute them by long division.

For √5, since 5 is between 4 (2²) and 9 (3²) and closer to 4, the root is a little above 2 — about 2.24. This estimation habit is priceless in multiple-choice aptitude tests.

Level 4: Decimal and Fraction Examples

Example: √1.44

Recognise that 1.2 × 1.2 = 1.44, so √1.44 = 1.2. Decimals often hide simple perfect squares.

Example: √(9/16)

Take the root of the top and bottom separately: √9 = 3 and √16 = 4, so √(9/16) = 3/4 = 0.75. Fractions become easy when both parts are perfect squares.

Benefits of Practising Examples

Graded examples turn an abstract idea into a skill. Working from easy perfect squares up to decimals teaches you to recognise which method fits which number, so in an exam you choose the fastest route automatically. Regular practice also cements the perfect squares in memory, which speeds up estimation and every later topic that uses roots, from the Pythagoras theorem to quadratic equations. For competitive-exam aspirants, this pattern recognition is the difference between solving a question in seconds and getting stuck.

Challenges and Limitations

Beginners often plateau when they meet non-perfect squares, because the tidy answers stop and decimals begin. Long division for these can feel laborious, and it is easy to make place-value slips. Fraction and decimal roots add another layer that trips up students who rush. The remedy is patience and gradual progression: master each level before moving to the next, and always verify by squaring your answer so an error never goes unnoticed.

Common Mistakes to Avoid

• Guessing instead of estimating. Use the nearest perfect squares to bracket your answer rather than guessing blindly.
• Forgetting to pair primes correctly. In factorisation, take only one number from each pair, not both.
• Mishandling decimals. Pair digits carefully around the decimal point in long division.
• Not checking by squaring. Always multiply your answer by itself to confirm it matches the original.
• Rounding too soon. Keep an extra decimal until the final step to stay accurate.
• Skipping levels. Jumping to hard examples before mastering perfect squares leads to confusion.

Best Practices and Expert Recommendations

• Progress level by level. Master perfect squares before tackling decimals and fractions.
• Recite the first 20 squares daily. Quick recall powers every other technique.
• Estimate before computing. Bracket each root between two perfect squares first.
• Verify by squaring. Confirm every answer by multiplying it by itself.
• Mix the methods. Practise factorisation, long division, and estimation so you can pick the best on sight.
• Check with a calculator. Confirm your worked examples and learn from any mismatch.

When an example involves dividing during long division, keep a division calculator nearby to verify each subtraction and quotient step.

Extra Practice Set with Full Solutions

The surest way to master square roots is deliberate practice, so here is an additional graded set with fully worked solutions written out in words. Attempt each one yourself before reading the answer, and try to name the method you would choose before you begin. This habit of choosing a method first is exactly what saves time in an exam.

Begin with the perfect square of two hundred and eighty-nine. Breaking it into prime factors gives seventeen multiplied by seventeen, a single pair, so the square root is simply seventeen. Next, take the perfect square of five hundred and seventy-six. Its prime factorisation is two multiplied by two, then two by two, then six by six in effect, and pairing the primes and taking one from each pair gives twenty-four as the root.

Now try a larger perfect square, one thousand and twenty-four. Recognising that thirty-two multiplied by thirty-two gives one thousand and twenty-four tells you the root is thirty-two, a value worth memorising because it appears often in problems involving powers of two. Move on to a number that is not a perfect square, such as ninety. It lies between eighty-one and one hundred, whose roots are nine and ten, and because ninety sits close to the middle, its root is roughly nine and a half, more precisely about nine point four nine.

For a decimal example, consider the square root of six point two five. Noticing that two point five multiplied by two point five gives six point two five reveals the root to be two point five exactly, a reminder that many decimals hide simple perfect squares. Finally, tackle a fraction: the square root of forty-nine over sixty-four. Taking the root of the top and the bottom separately gives seven over eight, which as a decimal is zero point eight seven five.

Working through a mixed set like this, where perfect squares, non-perfect squares, decimals, and fractions are jumbled together, trains you to identify the right technique quickly rather than mechanically applying one method to everything. That flexibility is the real skill that examiners reward, and regular practice with worked solutions is the fastest route to it.

Frequently Asked Questions

What are some easy square root examples for beginners?
Start with perfect squares: √4 = 2, √9 = 3, √16 = 4, √25 = 5, and √100 = 10. Each answer multiplied by itself gives the original number, which is the core idea of a square root.

How do I solve the square root of a large perfect square?
Use prime factorisation. Break the number into primes, group them in pairs, and take one number from each pair. For √324 you get 2 × 3 × 3 = 18.

What about numbers that are not perfect squares?
Their roots are decimals. Estimate by finding the two nearest perfect squares, or use the long division method for an exact value. For example, √10 is about 3.162.

How can I get better at square roots quickly?
Practise graded examples daily, memorise the first twenty perfect squares, and always check your answer by squaring it. Progressing level by level, from perfect squares to decimals, builds speed for exams.