Radicals & Rational Exponents — Print
A root is a fractional power. Once you accept that, every exponent rule you know still applies.
What the practice questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Rational — Any number writable as ab with integers a and b, b ≠ 0.
- Irrational — A real number that cannot be written as a fraction — its decimal never repeats and never ends.
- Factor — A number that divides evenly into another.
- Difference of squares — a2 - b2 = (a - b)(a + b).
- Rational exponent — x1/2 = √x and xm/n = √[n]xm.
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.
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 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.
- Largest perfect square dividing 200: 100. 200 = 100 × 2.
- √200 = √100√2 Split the root across the product.
- = 10√2 Check: 102 × 2 = 200 ✓
10√2
Adding radicals
Simplify √18 + √50.
- √18 = √9 · 2 = 3√2 Simplify each one first.
- √50 = √25 · 2 = 5√2 Same radicand now.
- 3√2 + 5√2 = 8√2 Add the coefficients, keep the radical.
- Not √68. Roots never add under one sign.
8√2
Rational exponent
Evaluate 274/3 without a calculator.
- The denominator 3 is the root: √[3]27 = 3 Take the root first to keep the numbers small.
- The numerator 4 is the power: 34 Now raise it.
- = 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.
Problem 1
Simplify √50.
Worked answer
5sqrt(2)
√25 · 2
= 5√2
Problem 2
Simplify √75.
Worked answer
5sqrt(3)
√25 · 3
= 5√3
Problem 3
Simplify √48.
Worked answer
4sqrt(3)
√16 · 3
= 4√3
Problem 4
Write √x using a rational exponent.
Worked answer
x^(1/2)
x1/2
Problem 5
Evaluate 161/2.
Worked answer
4
√16 = 4
Problem 6
Simplify √12 + √27.
Worked answer
5sqrt(3)
√12 = 2√3
√27 = 3√3
2√3 + 3√3 = 5√3
Problem 7
Simplify 3√5 - 7√5.
Worked answer
-4sqrt(5)
3 - 7 = -4
-4√5
Problem 8
Simplify √6 · √10.
Worked answer
2sqrt(15)
√60
= √4 · 15 = 2√15
Problem 9
Evaluate 272/3.
Worked answer
9
√[3]27 = 3
32 = 9
Problem 10
Evaluate 84/3.
Worked answer
16
√[3]8 = 2
24 = 16
Problem 11
Which sum is irrational?
- 3 + √2
- √4 + √9
- 12 + 13
- √16 - 1
Worked answer
3 + √2
√2 is irrational and 3 is rational
So the sum is irrational
All the others reduce to rationals
Problem 12
Write √[3]x5 using a rational exponent.
Worked answer
x^(5/3)
x5/3
Problem 13
Simplify √200 - √18.
Worked answer
7sqrt(2)
√200 = 10√2
√18 = 3√2
10√2 - 3√2 = 7√2
Problem 14
Expand (√3 + 2)(√3 - 2).
Worked answer
-1
(√3)2 - 22
= 3 - 4 = -1
A rational answer from irrational parts
Problem 15
Evaluate (116)-1/2.
Worked answer
4
(161)1/2
= √16 = 4
Problem 16
Is √8 · √2 rational or irrational?
Worked answer
rational
√16
= 4
Rational — two irrationals can multiply to a rational