§5.4, Irrational Numbers

Learning Targets I will define the irrational numbers Learning Targets I will define the irrational numbers. I will simplify square roots. I will perform operations with square roots. I will rationalize the denominator.

The Irrational Numbers The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating. For example, a well-known irrational number is π because there is no last digit in its decimal representation, and it is not a repeating decimal: π ≈ 3.1415926535897932384626433832795…

Square Roots The principal square root of a nonnegative number n, written , is the positive number that when multiplied by itself gives n. For example, because 6 · 6 = 36. Notice that is a rational number because 6 is a terminating decimal. Not all square roots are irrational.

Square Roots A perfect square is a number that is the square of a whole number. For example, here are a few perfect squares: 0 = 02 1 = 12 4 = 22 9 = 32 The square root of a perfect square is a whole number:

The Product Rule For Square Roots If a and b represent nonnegative numbers, then The square root of a product is the product of the square roots.

Example 1: Simplifying Square Roots Simplify, if possible: a. b. c. Because 17 has no perfect square factors (other than 1), it cannot be simplified.

Multiplying Square Roots If a and b are nonnegative, then we can use the product rule to multiply square roots. The product of the square roots is the square root of the product.

Example 2: Multiplying Square Roots Multiply: a. b. c.

Dividing Square Roots The Quotient Rule If a and b represent nonnegative real numbers and b ≠ 0, then The quotient of two square roots is the square root of the quotient.

Example 3: Dividing Square Roots Find the quotient: a. b.

Adding and Subtracting Square Roots The number that multiplies a square root is called the square root’s coefficient. Square roots with the same radicand can be added or subtracted by adding or subtracting their coefficients:

Example 4: Adding and Subtracting Square Roots Add or subtract as indicated: a. b. Solution:

Rationalizing the Denominator We rationalize the denominator to rewrite the expression so that the denominator no longer contains any radicals.

Example 6: Rationalizing Denominators Rationalize the denominator: a. b.

Homework Page 272, #18 – 40 (e), 50 – 70 (e)