If it is not a perfect square, you can still say exactly where it sits.
Estimate
Between which two consecutive whole numbers does √50 lie?
The perfect squares either side of 50 are 49 = 7² and 64 = 8², so √50 lies between 7 and 8 — and very close to 7, since 50 is only just past 49. D halves 50 instead of rooting it, which is the commonest confusion between "square root" and "half". A root is far smaller than half for any number above 4.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Knowing this row cold is most of the lesson. Every estimate below is just finding which two entries the number falls between. Past the table: 13² = 169, 14² = 196, 15² = 225 and 16² = 256.
A fraction is a perfect square when its top and bottom both are — root them separately.
The square root of a perfect square is rational. Every other whole number's root is irrational.
A screen showing 6.082762530 has simply run out of room. The digits carry on forever without ever repeating, which is exactly what makes √37 irrational.
√50
not a perfect square
49 < 50 < 64
the squares either side
7 < √50 < 8
root all three
The gap between consecutive squares widens as numbers grow, so a value just past a perfect square sits very close to that root — not halfway.
7.1² = 50.41
already past 50
7.0² = 49
below 50
50 is much nearer 50.41 than 49
so round up
√50 ≈ 7.1
true value 7.0710…
Squaring a candidate and comparing is always faster than trying to take a root by hand — and it proves the answer rather than hoping.
A square's side is the square root of its area.
A square vegetable plot has an area of 225 m². How long is each side, and how much fencing goes all the way round?
side = √225
area = side × side, so run it backwards
= 15 m
15 × 15 = 225
fencing = 4 × 15 = 60 m
four equal sides
Both signs: if x² = 196, what can x be?
14 × 14 = 196 and (−14) × (−14) = 196
two numbers square to 196
x = 14 or x = −14
√196 itself means only the positive one, 14
A length cannot be negative, so the plot's side is just 15. An equation like x² = 196 has no such limit, so it has two answers; if the question says x is negative, it is −14.
√16 + √9 = 4 + 3 = 7, but √16 + 9 = √25 = 5. Rooting each number and rooting the sum are different jobs, so a root of a sum must be added up first.
The same "what times itself" question in three less familiar dresses.
Find √0.49.
0.7 × 0.7 = 0.49
7 × 7 = 49, and two decimal places
√0.49 = 0.7
not 0.07
A root of a number between 0 and 1 is bigger than the number: 0.7 > 0.49.
Estimate √83 to the nearest tenth.
81 < 83 < 100, so 9 < √83 < 10
trap it
9.1² = 82.81, 9.2² = 84.64
83 is far nearer 82.81
√83 ≈ 9.1
just above 9
83 is 2 past 81 but 17 short of 100, so the answer hugs 9.
Find √2 14.
2 14 = 94
improper first
√94 = 32 = 1 12
root the top and the bottom
Rooting the parts separately, √2 and √14, would give an irrational number; convert first and it comes out exact.
Turn everything into the same kind of thing first.
Order: 5, √20, 4.5, √30
smallest first
√20 ≈ 4.5, since 4.5² = 20.25
and just under
√30 ≈ 5.5, since 5.5² = 30.25
and just under
√20 < 4.5 < 5 < √30
now comparable
Squaring everything works too: 25, 20, 20.25 and 30 put the same four in the same order, and avoids decimals entirely.
Expect a root to evaluate, a "between which two whole numbers", an ordering of roots and decimals, or a square area turned into a side. The fastest route is the squares table up to 15² and squaring every candidate; the 10-second check is to square your answer and see the original number come back.
Estimate to the nearest tenth
√30
25 < 30 < 36, so √30 is between 5 and 6. Testing: 5.5² = 30.25, just above 30, and 5.4² = 29.16, just below. 30 is nearer 30.25, so 5.5 to the nearest tenth. D halves 30 rather than rooting it.
Hint: find the two whole numbers first, then square one or two candidate tenths between them until you bracket 30.
