The square root of a number x is the number whose product with itself is x.
EXAMPLEIf you square the number -2, it equals 4. If you square the number 2, it also equals 4.
As you can see from the examples above, when you square any real number, the result will never be a negative number.
Since the square of a real number cannot be negative, the square root of a negative number must be a non-real number, otherwise known as an imaginary number. The imaginary unit, i, is defined as the square root of -1.
Imaginary numbers may be the result of solving a quadratic equation using the quadratic formula.
Imaginary numbers are written using the imaginary unit i in the form bi (b times i), where b is a real number. Recall that the product property for square roots states that for positive numbers a and b, the square root of a times b is equal to the square root of a times the square root of b:
You can also use the product property for square roots of negative numbers in the form bi.
EXAMPLEThe square root of -9 can be written as the square root of 9 times the square root of -1. The square root of 9 is 3, and the square root of -1 is defined as i. Therefore, you can write the square root of -9 as 3i.
Being able to identify perfect squares and appropriately using the product property for square roots is important when you are simplifying square roots and writing imaginary numbers. For example, suppose you want to simplify the expression:
You can start by simplifying in your parentheses.
Next, you square the 4, and subtract your terms, which equals -4.
Using the product property for square roots, you can rewrite the square root of -4 as the square root of 4 times the square root of -1. The square root of 4 is 2, and the square root of -1 is i, so your final answer is 2i.
Source: This work is adapted from Sophia author Colleen Atakpu.