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Square & Square Roots

About 7 min · Elementary Arithmetic · English

Time worked out from the lesson's 469 words at 150 words a minute, plus a minute for each question.

This topic tests your ability to identify perfect squares, extract square roots through standard methods, and apply these concepts to practical problems. Mastery here saves precious exam time because many percentage, average, and geometry problems depend on quick mental calculation of squares and roots.

Key ideas

• Perfect Square: A number obtained by multiplying an integer by itself. Examples: 1, 4, 9, 16, 25, 36... Perfect squares always end in 0, 1, 4, 5, 6, or 9; never in 2, 3, 7, or 8.
• Square Root: The inverse operation of squaring. If a² = b, then √b = a. Every positive number has two roots (positive and negative), but exam questions typically ask for the principal (positive) root.
• Prime Factorisation Method: Express the number as a product of prime factors. If all primes appear in pairs, the number is a perfect square. Take one factor from each pair to find the root.
• Long Division Method: A systematic algorithm for extracting square roots digit-by-digit, working left-to-right from the decimal point. Works for both perfect and non-perfect squares, producing decimal approximations when needed.
• Unit Digit Pattern: The last digit of a square depends only on the last digit of the original number: 1→1, 2→4, 3→9, 4→6, 5→5, 6→6, 7→9, 8→4, 9→1, 0→0. Use this to eliminate wrong answer choices quickly.

Watch out for

• Forgetting to pair digits from the right in long division → Incorrect grouping leads to wrong quotients. Always start pairing from the decimal point, moving outward. For 5329, group as 53|29, not 532|9.
• Assuming non-ending-in-square-digit means non-perfect-square → While 2, 3, 7, 8 endings guarantee non-perfect, a 0, 1, 4, 5, 6, 9 ending doesn't guarantee perfection. Example: 52 ends in 2 (non-perfect ✓), but 51 ends in 1 yet isn't perfect. Always verify with factorisation or long division.
• Missing pairs in prime factorisation → Students sometimes write √(2² × 3²) = 2 × 3 × √1 = 6√1 = 6 but forget to extract both factors. Correct approach: take one factor from each complete pair only.

Quick reference

• Squares 1–20 must be instant recall: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
• Perfect squares never end in 2, 3, 7, or 8.
• Prime factorisation method: pair up identical primes; if all pair, extract one from each pair.
• Long division: group pairs from decimal, find largest square ≤ first group, double quotient for next divisor.
• Sum of first n odd numbers = n².

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 17 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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