GCSE · Maths · AQA · Spec 8300 · Higher
Estimating powers and roots (Higher)
Is √40 closer to 6 or to 7? You don't need a calculator. You need two numbers you already know.
Watch it done
Problem
Without a calculator: (a) estimate √55 to 1 decimal place; (b) estimate ∛70 000.
AQA GCSE: this content can be assessed on any paper, including the non-calculator Paper 1, so write down the perfect powers you compared with.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Trap any root between two numbers you already know, then judge where it really sits.
What you need to know
- Square numbers up to 15 × 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
- Cube numbers: 1, 8, 27, 64, 125, and 1000 = 10³.
- Powers of 2, 3, 4 and 5, for example 2⁵ = 32, 3⁴ = 81, 4³ = 64 and 5³ = 125.
- 1000 = 10³ and 1 million = 10⁶, so √1 000 000 = 1000 and ∛1 000 000 = 100.
The big picture
To estimate a root without a calculator, trap the number between two perfect powers you know (squares, cubes, 1000 or 1 million), and its root is trapped between their roots. Then judge which end it's nearer, and check the halfway value when it's close. To estimate a power, round the base first and decide whether the true answer is a bit more or a bit less.
Key points
Worked example
Problem
Between which two whole numbers does the fourth root of 200 (∜200) lie, and which is it closer to?
⚠ Watch out
Halving instead of rooting: √64 is 8, not 32. Check by squaring back. 32² = 1024, nowhere near 64.
Memory hook
Trap it, judge it, check it: squeeze the number between two perfect powers, see which one it's nearer, then power up your answer to check.
Check yourself
Without a calculator, between which two whole numbers does √150 lie, and which is it closer to? Then roughly how big is 5.1³?
Flashcards
(9)Square numbers from 11² to 15²
The first five cube numbers, plus 10³
Powers of 2 up to 2¹⁰
Powers of 3, 4 and 5 to recognise
1000 and 1 million as powers of 10, and their roots
How do you trap a square root between two whole numbers?
When a number is near the middle, how do you tell which whole number its root is closer to?
Quick estimate of a power with a decimal base, like 3.1³
Is the square root of a number between 0 and 1 bigger or smaller than the number?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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