Powers & roots · Technique guide
Powers and Square Roots in Your Head
By Robert Hansson, Kung Fu Math
Squares and square roots appear everywhere: in areas, in Pythagoras, in statistics and in physics. Knowing the squares up to 20 by heart is a great start, but a handful of patterns let you go far beyond that without a calculator.
Each technique below turns a power or a root into a few small, friendly steps. Once you know them, numbers like 65², 49² or √200 stop being frightening and become a quick bit of thinking.
01
Square numbers ending in 5
To square a number ending in 5, take the digits in front of the 5, multiply them by the next whole number, and write 25 after the result.
For 65²: the front digit is 6, and 6 × 7 = 42. Write 25 after it: 4 225. This works because (10n + 5)² = 100·n(n + 1) + 25.
Worked example
85²
- → Front part: 8
- → 8 × 9 = 72
- → Write 25 after it: 7 225
Answer: 7225
Practice: Try 35², 75², 105² and 115².
02
Square near a round number
Use (a + b)² = a² + 2ab + b² with a as a round number. For 49², think (50 − 1)²: 2 500 − 100 + 1 = 2 401.
The middle term 2ab is the only real work, and with a round a it is easy.
Worked example
102²
- → 102 = 100 + 2
- → 100² = 10 000
- → 2 · 100 · 2 = 400
- → 2² = 4
- → 10 000 + 400 + 4 = 10 404
Answer: 10404
Practice: Try 31², 59², 98² and 201².
03
Find exact roots of perfect squares
If you know a number is a perfect square, two clues give the root. The last digit tells you the last digit of the root: a square ending in 6 has a root ending in 4 or 6. The size tells you the tens.
For √5 476: 70² = 4 900 and 80² = 6 400, so the root is in the seventies. It ends in 4 or 6. 75² = 5 625 is larger, so the root is 74.
Worked example
√2 209
- → 40² = 1 600 and 50² = 2 500, so the root is in the forties
- → Ends in 9, so the root ends in 3 or 7
- → 45² = 2 025 is smaller, so pick 47
Answer: 47
Practice: Try √1 444, √3 721 and √7 056.
04
Estimate any square root
For a number that is not a perfect square, find the nearest square below it. Add the leftover divided by twice the root. The estimate √(a² + r) ≈ a + r/(2a) is accurate to about two decimals for most everyday numbers.
Worked example
√52
- → Nearest square below: 49 = 7²
- → Leftover: 52 − 49 = 3
- → 3 ÷ (2 · 7) ≈ 0.21
- → √52 ≈ 7.21 (true value 7.211…)
Answer: ≈ 7.21
Practice: Estimate √27, √83 and √150.
05
Use the power laws
Powers obey simple rules: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ. They let you regroup a big power into small, known pieces.
For 2¹⁰ remember 1 024 ≈ one thousand. Then 2²⁰ ≈ one million and 2³⁰ ≈ one billion. For 5³ · 2³, group the pairs: (5 · 2)³ = 1 000.
Worked example
4⁵ · 25⁵
- → Group the bases: (4 · 25)⁵
- → 4 · 25 = 100
- → 100⁵ = 10 000 000 000
Answer: 10000000000
Practice: Try 2¹², 8 · 2⁷ and 20³ · 5³.
Common mistakes
- Writing 25 after n² instead of after n(n + 1) when squaring numbers ending in 5.
- Forgetting the middle term: (a + b)² is not a² + b².
- Multiplying exponents when you should add them: a² · a³ = a⁵, not a⁶.
- Thinking √(a + b) equals √a + √b. It does not.
Summary
- n5² = n(n + 1) followed by 25.
- Square near a round number with a² + 2ab + b².
- The last digit and the size pin down a perfect square root.
- √(a² + r) ≈ a + r/(2a) gives a quick estimate.
Into the dojo
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