Math Vale

🔢 Number Kingdom · Number Theory

Simplifying Radicals

Write a square root in simplest form by pulling out perfect-square factors, then add, multiply and divide radicals, and clear a root from a denominator.

In short

  • A root splits across a product, sqrt(p x q) = sqrt(p) x sqrt(q), and that single rule is what lets a perfect-square factor step outside the root sign.
  • A radical is in simplest form when the number underneath has no perfect square left as a factor, which is why you take the largest square you can find.
  • Roots multiply and divide straight across, with whole numbers outside handled in their own separate column.
  • Only like radicals add: 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2), and terms only become alike after each one has been simplified.
  • A finished answer carries no root in its denominator; multiplying top and bottom by that root clears it without changing the value.

A radical is a number, not a puzzle

sqrt(2) is a number. It sits between 1 and 2, closer to 1.4, and it goes on forever without repeating — so 1.414 is only a photograph of it, never the thing itself.

That is why mathematicians keep the root sign. sqrt(2) is exact; 1.414 is rounded. If a wall is sqrt(2) metres long and you build ten of them, the rounded version is already out by a centimetre.

Once you accept a root as a number, it obeys ordinary rules. In particular it can be multiplied by an ordinary whole number, and the shorthand for that is to write them side by side:

  • 3 sqrt(2) means 3 times sqrt(2), about 4.24.
  • sqrt(2) on its own means 1 sqrt(2) — the 1 is understood, exactly as it is in x.

The whole topic is about rewriting a root in the tidiest possible form, called simplest form, so that two people who solve the same problem write the same answer down.

Pulling perfect squares out

A root splits across a product. That single fact does all the work:

sqrt(p x q) = sqrt(p) x sqrt(q)

Test it: sqrt(4 x 9) = sqrt(36) = 6, and sqrt(4) x sqrt(9) = 2 x 3 = 6. The same.

So look inside a root for a factor that is a perfect square — 4, 9, 16, 25, 36, 49, 64, 81, 100 — because the root of that factor is a whole number and can step outside.

sqrt(50) = sqrt(25 x 2) = sqrt(25) x sqrt(2) = 5 sqrt(2)

A root is in simplest form when nothing under it divides by a perfect square any more. That is why you must take the *largest* perfect square you can find. Writing sqrt(72) as sqrt(4 x 18) = 2 sqrt(18) is true, but 18 still divides by 9, so you are only halfway; taking 36 instead gives 6 sqrt(2), and there it stops.

Warning: this splitting works for products only. sqrt(9 + 16) is sqrt(25) = 5, not 3 + 4 = 7. Roots do not split across a sum.

Multiplying and dividing radicals

Because roots split across products, they also join across them, which makes multiplication easy:

sqrt(6) x sqrt(10) = sqrt(60) = sqrt(4 x 15) = 2 sqrt(15)

Whole numbers outside multiply with each other and roots multiply with each other, in two separate columns:

2 sqrt(6) x 5 sqrt(10) = (2 x 5) sqrt(60) = 10 x 2 sqrt(15) = 20 sqrt(15)

Division follows the same pattern, with the numbers underneath dividing instead:

sqrt(96) / sqrt(2) = sqrt(48) = sqrt(16 x 3) = 4 sqrt(3)

Dividing first is nearly always the quicker route — it shrinks the number you have to factor. The product of two roots is almost never in simplest form already, so make simplifying the last step of every multiplication and division, not an optional extra.

Adding like radicals

Multiplying roots is easy; adding them is fussier, and for one reason: sqrt(a) + sqrt(b) is not sqrt(a + b). Check it once and you will never forget it — sqrt(9) + sqrt(16) = 3 + 4 = 7, while sqrt(25) is only 5.

Radicals add the way apples do. Three apples plus five apples is eight apples; 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2). The root is the *name* of the thing being counted, and only terms with the same name combine. Terms with the same number under the root are called like radicals.

So the recipe is: simplify every term first, then see which names match.

3 sqrt(2) + sqrt(50) = 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2)

Notice how much rides on that middle step. sqrt(50) does not contribute one sqrt(2), it contributes five, and forgetting that is the most common slip in the whole topic. And if the names never match — sqrt(2) + sqrt(3) — the expression is already as simple as it gets. Leave it alone.

Clearing a root from the denominator

By long tradition a final answer does not leave a root underneath a fraction. 1 / sqrt(2) and sqrt(2) / 2 are the same number, but only the second one is written down, because dividing by 1.414... is far harder by hand than halving.

Clearing it is called rationalising the denominator, and the trick is to multiply the top and the bottom by that same root. Multiplying by sqrt(b) / sqrt(b) is multiplying by 1, so the value cannot change — only its appearance does:

6 / sqrt(3) = (6 x sqrt(3)) / (sqrt(3) x sqrt(3)) = 6 sqrt(3) / 3 = 2 sqrt(3)

The bottom turns whole because sqrt(b) x sqrt(b) = b. Then reduce, exactly as you would reduce any fraction — the last step is easy to forget.

Radicals in simplest form are not housekeeping for its own sake. Solve a quadratic and the formula hands you a root of the discriminant, which almost always needs simplifying. Use the Pythagorean theorem on legs of 2 and 4 and the exact answer is sqrt(20) = 2 sqrt(5). Every one of those answers is a radical waiting to be tidied.

Worked examples

Example 1

Write sqrt(288) in simplest form.

  1. Look for the largest perfect square that divides 288. Try them in turn: 4 divides it, 9 divides it, and so does 144.
  2. 288 = 144 x 2, and 144 is a perfect square.
  3. sqrt(288) = sqrt(144) x sqrt(2) = 12 sqrt(2).
  4. Check that nothing is left: 2 does not divide by 4, 9 or 25, so 12 sqrt(2) is simplest form.

Example 2

Simplify 3 sqrt(2) + sqrt(50) - sqrt(8).

  1. Simplify each term before adding anything: sqrt(50) = sqrt(25 x 2) = 5 sqrt(2).
  2. sqrt(8) = sqrt(4 x 2) = 2 sqrt(2).
  3. Every term is now counted in sqrt(2)s: 3 + 5 - 2 = 6.
  4. So the answer is 6 sqrt(2). The number under the root never changes when terms are combined.

Example 3

Rationalise the denominator: 12 / sqrt(6).

  1. A root may not stay underneath, so multiply the top and the bottom by sqrt(6). That is multiplying by 1, so the value is untouched.
  2. The bottom becomes sqrt(6) x sqrt(6) = 6, and the top becomes 12 sqrt(6).
  3. That gives 12 sqrt(6) / 6.
  4. Reduce the fraction: 12 / 6 = 2, so the answer is 2 sqrt(6).

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Write sqrt(63) in simplest form. A root is in simplest form when the number underneath has no perfect square left as a factor. a sqrt(b) means a times sqrt(b). Answer as the pair (a, b): the whole number outside the root first, the number left under the root second.

Answer: (3, 7)

  1. 63 = 9 x 7, and 9 is a perfect square.
  2. sqrt(63) = sqrt(9) x sqrt(7) = 3 sqrt(7).
  3. 7 has no perfect-square factor left, so the simplest form is 3 sqrt(7), written (3, 7).

Problem 2

Difficulty 3 of 5

Multiply and simplify: 3 sqrt(65) x sqrt(5) a sqrt(b) means a times sqrt(b). Answer as the pair (a, b): the whole number outside the root first, the number left under the root second.

Answer: (15, 13)

  1. sqrt(65) x sqrt(5) = sqrt(65 x 5) = sqrt(325)
  2. The whole numbers outside multiply too: 3 x 1 = 3.
  3. 325 = 25 x 13, so sqrt(325) = 5 sqrt(13).
  4. Altogether that is 15 sqrt(13), written (15, 13).

Problem 3

Difficulty 4 of 5

Simplify: 2 sqrt(208) + 5 sqrt(325) a sqrt(b) means a times sqrt(b). Answer as the pair (a, b): the whole number outside the root first, the number left under the root second.

Answer: (33, 13)

  1. sqrt(208) = sqrt(16 x 13) = 4 sqrt(13), so 2 sqrt(208) = 8 sqrt(13).
  2. sqrt(325) = sqrt(25 x 13) = 5 sqrt(13), so 5 sqrt(325) = 25 sqrt(13).
  3. Every term is now counted in sqrt(13)s: 8 + 25 = 33.
  4. So the answer is 33 sqrt(13), written (33, 13).

Common mistakes

  • Stopping halfway: answering 2 sqrt(18) for sqrt(72). It is equal, but 18 still divides by 9, so 6 sqrt(2) is the simplest form.
  • Halving instead of rooting: reading sqrt(72) as 36. A root shrinks a number far more than halving does.
  • Adding the numbers under the roots: writing sqrt(9) + sqrt(16) as sqrt(25). That is 5, but the real answer is 3 + 4 = 7.
  • Forgetting the new coefficient after simplifying: adding 3 sqrt(2) + sqrt(50) as 4 sqrt(2), when sqrt(50) is worth five sqrt(2)s, not one.
  • Leaving the root in the denominator, or clearing it and then forgetting to reduce: 6 sqrt(3) / 3 still simplifies to 2 sqrt(3).
  • Cancelling the root away: treating 6 / sqrt(3) as 6 / 3 = 2. sqrt(3) is about 1.73, not 3, so the answer is nearer 3.46.

What you should be able to do

  • Simplify a square root such as sqrt(50) to 5 sqrt(2) by factoring out the largest perfect square.
  • Multiply and divide square roots, and simplify the product.
  • Add and subtract like radicals after simplifying each one.
  • Rationalise a denominator such as 6 / sqrt(3).

Where this fits in the curriculum

Common Core

  • HSN-RN.A.2

    High school — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

  • 8.EE.A.2

    Grade 8 — Use square root and cube root symbols to represent solutions to equations, and know that sqrt(2) is irrational.

Ontario

  • G8.B1.3

    Grade 8 — Estimate and calculate square roots, in various contexts.

    Ontario has no expectation for writing a square root in simplest radical form before Grade 10 (MPM2D); Grade 8 square-root work is the nearest, so this pairing is a judgement call.

SAT

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