How to use
- Type the number and pick the root: square, cube, 4th, 5th or any n.
- Read the decimal result, the exact simplified radical and the prime-factor steps.
- Scroll down for a 20-question worksheet; change the worksheet number for a new set and print it.
Worked example: The 4th root of 48
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3. Groups of four come out of the radical: one group of four 2s gives a 2 outside, leaving 3 inside. The exact form is 2 ∜3, and the decimal is 2.6321.
For √72: 72 = 2³ × 3². Pairs come out: one pair of 2s and one pair of 3s give 2 × 3 = 6 outside, leaving a single 2 inside. √72 = 6√2 ≈ 8.4853.
Frequently asked questions
What does the exact form mean?
A radical written as k × ⁿ√r where r has no factor that is a perfect nth power. √72 = 6√2. It is found by prime factorisation and grouping.
How do odd roots of negative numbers work?
An odd root keeps the sign: ∛(−27) = −3, because (−3)³ = −27.
Why does √(−16) show 4i?
No real number squares to a negative. The even root of a negative is imaginary, written with i where i² = −1. √(−16) = 4i.
How is a perfect square detected?
If every prime factor appears an even number of times, the root is a whole number. 144 = 2⁴ × 3², so √144 = 12.
How accurate is the decimal?
Square roots use the exact machine square root; cube roots use the exact cube root function; other indices use exponentiation. Display precision is adjustable up to 10 decimals.
How do I make a different worksheet?
Change the worksheet number. Each number produces the same set every time, so you can print the questions and answers separately or share the number with a class.