Simplifying a root means walking every perfect-square factor out the door: √12 = √4·3 = 2√3.
Up to now a square root has been a question — “√12, what side gives an area of 12?” In this lesson the answer becomes a thing you can hold in your hand: √12 is just a number, exactly equal to 2√3, and like any number you can simplify it, multiply it, add it to its twins, and tidy it out of a denominator. By the end you'll move roots around as comfortably as you move whole numbers.
Keep one steady color habit in view all lesson. A radicand — the number living under the root sign — and the whole √ expression are teal. A perfect-square factor you pull out front is amber. A simplified or combined result is purple. A denominator is blue.
6.5.1 What a square-root expression means, and when it's valid
Write √a and you have named a single number: the non-negative number whose square isa. So √25 isn't an unfinished problem — it is5, because 5 × 5 = 25. The radical sign is a finished noun, not a verb waiting to run.
Some radicands come out clean. When a is a perfect square — 1, 4, 9, 16, 25, 36, … — the root is a whole number you can write down. Most radicands don't: √7 has no fraction or terminating decimal equal to it (you met that fact in 6.4 — it's irrational). That's fine. We simply leave it as √7: an exact number that sits between 2 and 3, never to be “finished” as a decimal.
There is one rule the radicand must obey. A square is never negative, so “the number whose square is a” can only exist when a ≥ 0. Ask for √−4 and no real number answers — nothing real squares to a negative. A real square root needs a radicand that is zero or positive.
The radicand line. To the left of 0 the root is undefined; a perfect square gives a whole-number root; everything else gives an exact root we leave under the sign.
Watch out
“√9” means the principal (non-negative) root, so √9 = 3 only — not ±3. The equation x² = 9 has two answers, x = 3 or x = −3, but the symbol √9 by itself always points to the positive one.
🎮 Try itIs this root valid — and does it come out clean?
Slide the radicand a. Watch where it becomes undefined, where it lands on a perfect square, and where it stays exact.
Radicand a12
6.5.2 Simplest form, and how to simplify a root
Here is the engine that powers this whole lesson. If a radicand hides a perfect-square factor, that factor can step out from under the root as its own square root. The reason is the rule √a·b = √a · √b. So
√12 = √4·3 = √4 · √3 = 2 · √3 = 2√3.
The amber 4 was a perfect square hiding inside 12; its root, 2, walks out front, and the 3 that has no square factor stays behind. A root is in simplest form when no perfect-square factor (other than 1) is left under the sign. The fast method: find the largest perfect-square factor and pull it out in one move.
An area of 12 splits as 4×3. The perfect-square 4 contributes a side of 2 that leaves the root; the leftover 3 stays as √3.
Worked example · pulling out the largest square
√50: the largest square factor of 50 is 25, and 50 = 25·2. So √50 = √25·√2 = 5√2.
√72: 72 = 36·2, so √72 = 6√2. (Miss the largest square and take 72 = 4·18 instead? You'd get 2√18, then have to simplify √18 = 3√2 again — same answer, more steps.)
Watch out
√a+b is not√a + √b. The split rule only works across multiplication. Check: √9+16 = √25 = 5, but √9 + √16 = 3 + 4 = 7. Different! Never break a sum apart under a root.
🎮 Try itThe simplify machine
Pick a radicand n. The machine factors out the largest perfect square and shows the simplest form.
n12
6.5.3 Multiplying and dividing roots
Two roots multiply by simply joining their radicands under one sign, and divide by dividing under one sign:
√a·√b = √a·b
and
√a√b = √a/b
Both hold for a, b ≥ 0 (and for division b > 0, since you can't divide by zero). After you join the radicands, always simplify the result. Two examples that come out beautifully clean:
√2·√8 = √16 = 4√18√2 = √9 = 3
Multiplying pours both radicands into one root: √2·√8 = √16 = 4. Why it works: (√2·√8)² = 2·8 = 16.
Worked example · join, then simplify
√6·√3 = √18 = √9·√2 = 3√2. Join first (6·3 = 18), then pull out the square (9).
A coefficient out front just multiplies along: 2√3 · 5√2 = (2·5)·√3·2 = 10√6.
🎮 Try itJoin two roots under one sign
Choose a and b, pick × or ÷, and watch them merge into one root — then simplify.
a2
b8
op
6.5.4 Adding and subtracting like roots
Adding roots is exactly like collecting like terms. Think of √3 as a single unit — call it “one root-three.” Then 2√3 means two of them, and
2√3 + √3 = 3√3
(two root-threes plus one root-three makes three root-threes)
You add the counts out front; the root itself rides along unchanged, just as 2 apples + 1 apple = 3 apples. But the units must match. 2√3 + √2will not combine — root-three and root-two are different units, like apples and oranges. You just leave it as 2√3 + √2.
Sometimes two roots look unlike but become like roots after you simplify. That's why simplifying first pays off:
√12 + √3 = 2√3 + √3 = 3√3
Like roots stack: two √3-lengths plus one make 3√3. A √2-length is a different unit and won't join the pile.
Watch out
You may not add the radicands: √2 + √3 ≠ √5. Check with decimals: 1.41 + 1.73 ≈ 3.14, but √5 ≈ 2.24. Addition lives outside the root, never inside it.
🎮 Try itThe like-roots adder
Build p√m + q√n. If the roots match it combines; if not, it tells you why. Try the “simplify first” case.
p2
√m3
q1
√n3
6.5.5 Rationalizing the denominator
A root sitting in a denominator feels awkward — it's hard to picture 1√2, dividing by a never-ending decimal. The fix is a clever form of multiplying by 1: multiply top and bottom by that same root. The denominator √2·√2 becomes √4 = 2 — a whole number — and the root moves upstairs, out of the basement:
1√2
= 1√2 · √2√2
= √22
= √22
The value never changed — 1√2 and √22 are both about 0.707 — but the second is the tidy, standard form, with no root left below the bar.
Multiplying by √2√2 (which is just 1) lifts the root from the denominator into the numerator and leaves a whole number below.
Worked example · sometimes it simplifies all the way
3√3 = 3√3·√3√3 = 3√33 = √3. The 3 on top and the 3 on the bottom cancel, leaving just √3.
🎮 Try itThe rationalize machine
Pick a numerator c and a denominator root √d. Watch the root climb upstairs, step by step.
c1
√d2
6.5.6 Mixed operations with square-root expressions
Roots obey the same order of operations as ordinary numbers: do powers and roots first, then × and ÷, then + and − last. Treat each root as a number and march through in order. Here is a full one worked end to end:
(√2)² + √3·√12 − √8
Step 1 — powers/roots: (√2)² = 2 (squaring undoes the root), and √8 = 2√2. Step 2 — multiply: √3·√12 = √36 = 6. Step 3 — add/subtract left to right: 2 + 6 − 2√2 = 8 − 2√2. The whole-number parts combine; the leftover root rides along.
The same priority ladder as always — powers/roots, then ×÷, then +−. Each rung turns roots into plain numbers until only 8 − 2√2 remains.
Watch out
(√2)² = 2, but don't confuse it with √2² = √4 = 2 — same answer here only because squaring and rooting are inverses. And the final 8 − 2√2 is done: the 8 and the 2√2 are unlike (a whole number versus a root) and cannot merge into “6√2” or anything else.
🎮 Try itThe mixed-operations stepper
Step through (√2)² + √3·√12 − √8 one priority level at a time.
★ The big ideas, in one breath
A square root is a number: √a exists exactly when a ≥ 0, equals a whole number when a is a perfect square, and is otherwise an exact value we leave under the sign. Tidy it by walking out every perfect-square factor (√12 = 2√3). Multiply and divide by joining radicands under one root (√a·√b = √ab); add and subtract only like roots, counting them like terms; and lift a root out of a denominator by multiplying top and bottom by it. Through it all, addition stays outside the root and the principal root stays non-negative.
Coming up next — 6.6
Now that you can compute with roots exactly, lesson 6.6 · Operating with & Estimating Reals places them on the number line: how big is 8 − 2√2, really? You'll pin exact roots between fractions, estimate to a decimal, and operate on rationals and irrationals together — the gateway into algebra in Stage 7.
✎ Exercises 6.5
Work each one out first, then open the answer to check your thinking.
Evaluate √36. Is the radicand a perfect square?
Show answer
√36 = 6, because 6² = 36. Yes, 36 is a perfect square, so the root comes out as a whole number.
Which of these is not a real number, and why? √0, √7, √−9.
Show answer
√−9 is not a real number: no real number squares to −9. The other two are fine — √0 = 0, and √7 is an exact (irrational) number between 2 and 3.
Simplify √18.
Show answer
18 = 9·2, so √18 = √9·√2 = 3√2.
Simplify √75.
Show answer
The largest square factor of 75 is 25: 75 = 25·3. So √75 = 5√3.
Compute √3·√27.
Show answer
Join the radicands: √3·27 = √81 = 9. (Check: 81 is 9², a perfect square.)
Compute √50√2.
Show answer
Divide under one root: √50/2 = √25 = 5.
Add 5√2 + 3√2. Then explain why 5√2 + 3√5 cannot be combined.
Show answer
Same unit (root-two): (5 + 3)√2 = 8√2. But √2 and √5 are different units, like apples and oranges, so 5√2 + 3√5 stays as it is.
Simplify first, then add: √8 + √18.
Show answer
√8 = 2√2 and √18 = 3√2. Now they're like roots: 2√2 + 3√2 = 5√2.
Rationalize the denominator of 6√3.
Show answer
Multiply by √3√3: 6√33 = 2√3 (the 6 over 3 reduces to 2).
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The anchor standard is 8.EE.A.2 — using square-root symbols to represent solutions and evaluating the roots of small perfect squares — and the lesson treats a root as a definite number rather than an unfinished operation. The simplifying, multiplying/dividing, adding like roots, and rationalizing here are an early, concrete preview of high-school N-RN.2 (rewriting radical expressions) and N-RN.3 (sums and products of rational and irrational numbers), kept deliberately numerical so it's reachable in middle school.
The #1 misconception is splitting a root across addition: writing √a+b = √a + √b (so “√2 + √3 = √5”). Antidote: have them test once with numbers — √9+16 = 5 but √9 + √16 = 7 — and contrast it with the rule that does split, √ab = √a·√b. Roots split over multiplication, never over addition. (Watch also for “√9 = ±3”: the symbol means the principal, non-negative root.)