Writing a surd as a single root and comparing surds
Which number is bigger: 2√7 or 3√3? The first one has a smaller number in front of the root but a bigger one under it. We can decide without a calculator using a simple trick: we rewrite both numbers so that everything is under the root.
This step is the opposite of simplifying a surd. We move the number in front of the root under the root. (In the US, surds are called radicals.)
Every number that is not negative can be written as the square root of its square: 3 = √9, because 3² = 9. So:
a√b = √(a² · b) (for a ≥ 0, b ≥ 0)
We square the number in front of the root and multiply the number under the root by it.
For a cube root we cube the number instead: a∛b = ∛(a³ · b).
Square roots
Worked example 1
3√2
3 = √9.
3√2 = √9 · √2 = √(9 · 2) = √18.
Check by simplifying: √18 = √(9 · 2) = 3√2 ✓.
Worked example 2
5√3
5² = 25.
5√3 = √(25 · 3) = √75.
Cube roots and fourth roots
Worked example 3
2∛5
For a cube root we cube the number: 2³ = 8.
2∛5 = ∛(8 · 5) = ∛40.
Worked example 4
3∜2 (a fourth root, so we raise the number to the power 4)
3⁴ = 81.
3∜2 = ∜(81 · 2) = ∜162.
Comparing surds
The square root keeps growing: the bigger the number under it, the bigger the root. So we just move both numbers under the root and compare the numbers under it.
Worked example 5
2√7 and 3√3
2√7 = √(4 · 7) = √28.
3√3 = √(9 · 3) = √27.
28 > 27, so 2√7 > 3√3.
A calculator confirms it: 2√7 ≈ 5.29 and 3√3 ≈ 5.20.
Worked example 6
3√5 and 7
3√5 = √45.
7 = √49.
45 < 49, so 3√5 < 7.
Worked example 7
2√18 and 3√8
2√18 = √(4 · 18) = √72.
3√8 = √(9 · 8) = √72.
The two numbers are equal: 2√18 = 3√8. (Simplified, both are 6√2.)
Worked example 8
3∛2 and 2∛6
3∛2 = ∛(27 · 2) = ∛54.
2∛6 = ∛(8 · 6) = ∛48.
54 > 48, so 3∛2 > 2∛6.
Common mistakes
Forgetting to square the number. 3√2 is not √6. The 3 goes under the root as 3² = 9, which gives √18.
Squaring for a cube root. 2∛5 is not ∛20. The correct way is 2³ · 5 = 40, which gives ∛40.
Comparing only the numbers in front. 3√3 has a bigger number in front than 2√7, and yet it is smaller.
Comparing different roots. You cannot compare √28 and ∛28 this way. Both numbers must be under the same kind of root.
Summary
a√b = √(a²b), a∛b = ∛(a³b).
Moving a number under the root is the opposite of simplifying a surd.
To compare two surds, rewrite both as a single root of the same kind and compare the numbers under them.
A whole number can be written as a root too: 7 = √49.
