Lesson 62: Square Roots, Higher Order Roots, Evaluating Using Plus or Minus
Every positive number has one positive and one negative square root. For example, the number 16 has two square roots: 4 and -4. we can verify that 4 and -4 are square roots of 16 by multiplying each by itself to get 16. (4)(4) = 16 (-4)(-4) = 16
We use the square root radical sign, √, to designate the positive square root. Therefor, √4 = 2 since 2 squared equals 4. If we wish to indicate the negative square root of 4, we must write -√4 = -2
Definition of Square Root: If x is greater than zero, then √x is the unique positive real number such that (√x) = x 2
Using this definition, we can say that √2√2 = 2 √a√a = a √2.42√2.42 = 2.42 √x – 1√x – 1 = x – 1
Rational numbers can be defined as a number whose decimal representation is a non-repeating decimal numeral of infinite length.
1/7 = 0.142857142857 142857 repeats itself so 1/7 is not rational. √2 however, is a rational number because the numbers never repeat and go on forever.
We can use a calculator to give us a decimal approximation, however, we will never be able to write an exact decimal representation of an irrational number.
Example: Use a calculator to determine to five decimal places √18.
The notion of square roots can be extended. For example, when we refer to the cube root of a given number, we are referring to the number that when multiplied by itself three times produces the given number. It turns out that every real number only has one real number cube root, which is either negative or positive.
For example, √8 = 2 since 2 = 8 √-8 = -2 since (-2) = -8 3
Definition of Cube Root: If x is a real number, then √x is the unique positive or negative real number such that (√x) = x. 3 3
Fourth roots can be defined similarly. However, like square roots, there are two fourth real number roots of a real number – one positive and one negative. When we use the radical sign to refer to a fourth root, we are referring to the positive fourth root.
Definition of Fourth Root: If x is a real number, then √x is the unique positive real number such that (√x) = x. 4 4
Practice: Without using a calculator, determine the following. Note that all answers are integers. a) √8 b) √81 c) √-27 343343
Answer: a) (2)(2)(2) = 8 b) (3)(3)(3)(3) = 81 c) (-3)(-3)(-3) = -27
The equation x = 4 has two solutions. 2 and -2. we could write x = 2 and x = -2 or we could write x = ± 2. 2