A root of a number is a value that, when multiplied by itself a certain number of times, gives the original number. The nth root of x is the number that, when raised to the power n, equals x.
The most common roots are the square root (2nd root) and cube root (3rd root). Square roots are used extensively in geometry, physics, and statistics, while cube roots appear in volume calculations and engineering.
The nth root of x, written as nāx or x^(1/n), can be calculated as:
For example, the square root of 9 is 3 because 3 Ć 3 = 9. The cube root of 8 is 2 because 2 Ć 2 Ć 2 = 8.
Key Properties:
Problem: Calculate ā144
Solution:
Problem: Calculate ā27
Solution:
Problem: Calculate ā“ā81
Solution:
The square root of a negative number is an imaginary number. For example, ā(-1) = i, where i is the imaginary unit. The square root of -4 is 2i. This calculator returns NaN for even roots of negative numbers.
āxĀ² = |x| (always positive, the absolute value of x), while (āx)Ā² = x (only defined for x ā„ 0, equals x). The order of operations matters!
Yes! Odd roots of negative numbers are defined and result in negative numbers. For example, ā(-8) = -2 because (-2)Ā³ = -8.
To simplify āx, factor out perfect squares. For example, ā72 = ā(36 Ć 2) = 6ā2. Look for the largest perfect square that divides evenly into the number under the radical.