Radicals & Rational Exponents

A root is a fractional power. Once you accept that, every exponent rule you know still applies.

You have 72 square tiles and want to lay them out as a perfect square patio.

It cannot be done. 8 × 8 is 64 and 9 × 9 is 81 — nothing lands on 72.

a square patio of 72 tiles — how long is one side? 6 6 6 36 tiles 36 tiles 72 tiles will not make a square — but 36 will, twice over

So the side is √72, somewhere between 8 and 9. Can that be written more usefully?

Split the 72 into 36 and 2. 36 is a perfect square, so √36 = 6 comes straight out.

That leaves √72 = 6√2 — one exact number, about 8.49.

Pulling out the 36 is not a trick: it is the largest square hiding inside 72. Use 4 instead and you get 2√18 — true, but not finished, because 18 still has a 9 in it.

Simplifying a square root by pulling out a perfect square pull out the largest perfect square sqrt 72 sqrt(36 . 2) sqrt36 . sqrt2 6 sqrt2 start square factor split the root done 36 is the biggest perfect square dividing 72; check it: 6^2 . 2 = 36 . 2 = 72 using 4 instead gives 2 sqrt18, which is right but not yet finished

Simplifying a root

Find the largest perfect square factor of the radicand, split the root, and take that square out.

√72 = √36 · 2 = 6√2.

If you use a smaller square, you are not wrong — just not finished. √72 = 2√18 needs another round.

Know your squares

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Check the radicand against that list from the biggest down, and the first hit is the one you want.

Adding and multiplying

You may only add radicals with the same radicand: 3√5 + 4√5 = 7√5. Treat √5 like a variable.

√2 + √3 cannot be combined at all. It is certainly not √5.

Multiplying is easier: for real a,b ≥ 0, √a · √b = √ab, then simplify.

Roots do not add

√9 + √16 = 3 + 4 = 7, but √25 = 5. So splitting a square root across a sum is not a general rule. For a,b ≥ 0, equality holds only when at least one of them is zero.

Rational exponents

x1/2 means √x; x1/3 means √[3]x. The denominator of the exponent is the root.

x2/3 means √[3]x2, or equivalently (√[3]x)2. Take the root first when you can — the numbers stay small.

82/3 = (√[3]8)2 = 22 = 4. Much easier than squaring 8 first.

Denominator is the root

In xm/n, the bottom n is the root and the top m is the power. Root on the bottom — it is the only thing to remember here.

Rational or irrational?

The square root of a perfect square is rational. Every other square root of a whole number is irrational.

Rational + irrational is always irrational. Rational × irrational is irrational, unless the rational is zero.

Irrational + irrational can go either way: √2 + (-√2) = 0, which is rational.

Closure

The rationals are closed under +, -, × and ÷ (except by zero). The irrationals are closed under none of them.

Worked examples

Simplest radical form

Simplify √200.

  1. Largest perfect square dividing 200: 100. 200 = 100 × 2.
  2. √200 = √100√2 Split the root across the product.
  3. = 10√2 Check: 102 × 2 = 200 ✓

10√2

Adding radicals

Simplify √18 + √50.

  1. √18 = √9 · 2 = 3√2 Simplify each one first.
  2. √50 = √25 · 2 = 5√2 Same radicand now.
  3. 3√2 + 5√2 = 8√2 Add the coefficients, keep the radical.
  4. Not √68. Roots never add under one sign.

8√2

Rational exponent

Evaluate 274/3 without a calculator.

  1. The denominator 3 is the root: √[3]27 = 3 Take the root first to keep the numbers small.
  2. The numerator 4 is the power: 34 Now raise it.
  3. = 81 The other order would mean √[3]531441 — same answer, far more work.

81

Words to know

Radical
A root expression, like √50.
Radicand
The thing under the root sign.
Simplest radical form
No perfect square factors left under the root.
Rational exponent
x1/2 = √x and xm/n = √[n]xm.
Rationalise
Clear a radical out of a denominator.

Mistakes to watch for

Writing √a + √b as √a+b. Test it: √9 + √16 = 3 + 4 = 7, but √25 = 5. Roots simply do not add that way — though they do multiply that way.

Using a small perfect square and stopping. √72 = 2√18 is correct but unfinished. Work down from the largest square that divides — 144, 121, 100, 81, 64, 49, 36 … — and the first hit is the right one.

Reading the numerator of a fractional exponent as the root. In xm/n the denominator is the root and the numerator the power. For 272/3: cube root first, giving 3, then square it.

Adding radicals with different radicands. 3√2 + 4√5 cannot be combined, exactly as 3a + 4b cannot. Treat √2 like a variable and the rule becomes obvious.

The Pythagoreans believed every length was a ratio of whole numbers. Then one of them proved √2 is not.

Legend says the discoverer, Hippasus, was drowned at sea for it — though the story may well have been invented later.

The proof survives and is short enough to fit on a postcard. It is usually the first proof by contradiction anyone meets.

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