Squares & Square Roots
Squaring and taking a square root undo each other. A root asks: what number, times itself, gives this?
You have 16 square tiles and want to lay them in a perfect square. Four rows of four. Easy.
Now try it with 20 tiles.
Four by four uses 16 and leaves four over. Five by five needs 25 and you are five short.
There is no whole number that works. The side of a 20-tile square is somewhere between 4 and 5.
That side length has a name: √20. It is a real length — you could cut it with scissors — it is just not a whole number.
So √n asks: what side length gives a square of area n? Sixteen answers neatly, twenty does not, and that difference is what this unit is about.
Squares and roots are opposites
72 = 49, so √49 = 7. They undo each other for nonnegative inputs. For any real x, √x2 = |x|; squaring loses the original sign.
Know these by heart — they turn up constantly: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
Recognising 144 as 122 on sight saves time in every later unit, especially Pythagoras.
Why "square"?
A square with side 7 has area 49. Going from side to area is squaring; going from area back to side is taking the root. The words are literal.
Estimating roots that are not exact
√40 is not a whole number. Trap it between the perfect squares either side: 36 < 40 < 49, so 6 < √40 < 7.
40 is much closer to 36 than to 49, so the answer is a little above 6 — about 6.3.
This bracket-and-judge method is usually all you need, and it catches calculator slips.
Simplifying a radical
Pull out any perfect-square factor. √72 = √36 × 2 = √36 × √2 = 6√2.
Look for the largest perfect square that divides the number. Using 4 instead of 36 still works, it just takes another round.
For a,b ≥ 0, √a × √b = √ab — real square roots multiply straight across, which is what makes this legal.
Solving x2 = k
x2 = 25 has two answers: x = 5 and x = -5, because both square to 25.
But the symbol √25 means only the nonnegative one, 5. That is a convention, and it trips people up.
So: solving an equation gives ±; evaluating a square root gives the nonnegative root. If k=0, x2=k has one real solution; if k<0, it has no real solution.
Two different questions
"Solve x2 = 16" → x = ± 4.
"Find √16" → 4.
Read carefully which one is being asked.
Roots of fractions and decimals
√925 = √9√25 = 35 — take the root of the top and the bottom separately.
√0.04 = 0.2, because 0.2 × 0.2 = 0.04. Note the root of a number below 1 is larger than the number.
Worked examples
Estimate
Between which two consecutive whole numbers does √130 lie, and which is it nearer?
- Find the perfect squares either side: 121 and 144. 112 = 121 and 122 = 144.
- 121 < 130 < 144, so 11 < √130 < 12 Bigger radicand means bigger root.
- 130 - 121 = 9 but 144 - 130 = 14 It sits closer to 121.
Between 11 and 12, nearer 11 — about 11.4.
Simplify
Simplify √98.
- Look for a perfect-square factor of 98. 98 = 49 × 2, and 49 is a perfect square.
- √98 = √49 × √2 Roots split across multiplication.
- = 7√2 √2 will not simplify further — 2 has no square factors.
7√2
Solve versus evaluate
Solve x2 = 81, then find √81.
- For the equation, ask what squares to 81. Both 9 and -9 do.
- x = 9 or x = -9, written x = ± 9 An equation keeps both solutions.
- But √81 = 9 only. The radical symbol means the positive root by definition.
x = ± 9, but √81 = 9.
Words to know
- Perfect square
- A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root
- For k ≥ 0, √k is the nonnegative number whose square is k.
- Radical
- The √ symbol, and the expression under it.
- Radicand
- The number under the radical sign.
Mistakes to watch for
Halving instead of rooting. √36 is 6, not 18 — you want the number that multiplies by itself to give 36, not the one that adds to itself. Check by squaring your answer.
Giving only the positive answer when solving x2 = k. Both 6 and -6 square to 36, so x2 = 36 has two solutions. (The symbol √36 on its own does mean just 6 — the two situations are different.)
Writing √a + b = √a + √b. Try a = 9, b = 16: the left is √25 = 5, the right is 3 + 4 = 7. Roots split across multiplication, never across addition.
Assuming a root must be irrational. √49, √0.25 and √49 are all perfectly ordinary rational numbers.
The radical sign √ is thought to be a stylised letter r, for the Latin radix, "root" — the same root as radish and radical.
It first appears in a 1525 German algebra book by Christoff Rudolff. The horizontal bar over the top came later, from Descartes.
Before that, writers just wrote the word. The symbol saved so much space that it spread across Europe within a century.