Index Radicand When the index of the radical is not shown then it is understood to be an index of 2 Radical
EXAMPLE 1: a)Give 4 examples of radicals b)Use a different radicand and index for each radical c)Explain the meaning of the index of each radical
Evaluate each radical: = 0.5 = 6 = 2 = 5 EXAMPLE 2:
4.2 Irrational Numbers
WORK WITH YOUR PARTNER 1. How are radicals that are rational numbers different from radicals that are not rational numbers? Rational Numbers: These are rational numbers: These are NOT rational numbers:
2. Which of these radicals are rational numbers? Which ones are not rational numbers? How do you know? WORK WITH YOUR PARTNER
RATIONAL NUMBERS a.Can be written in the form b.Radicals that are square roots of perfect squares, cube roots of perfect cubes etc.. c.They have decimal representation which terminate or repeats
IRATIONAL NUMBERS a.Can not be written in the form b.They are non-repeating and non-terminating decimals
EXAMPLE 1: Tell whether each number is rational or irrational. Explain how do you know. Rational, because 8/27 is a perfect cube. Also, 2/3 or 0.666… is a repeating decimal. Irrational, because 14 is not a perfect square. Also, √14 is NOT a repeating decimal and DOES NOT terminate Rational, because 0.5 terminates. Irrational, because π is not a repeating decimal and does not terminates
POWER POINT PRACTICE PROBLEM Tell whether each number is rational or irrational. Explain how do you know.
EXAMPLE 2: Use a number line to order these numbers from least to greatest Use Calculators!
EXAMPLE 2: Use a number line to order these numbers from least to greatest
POWERPOINT PRACTICE PROBLEM Use a number line to order these numbers from least to greatest
HOMEWORK O PAGES: O PROBLEMS: 3 – 6, 9, 15, 20, 18,
4.3 Mixed and Entire Radicals
Index Radicand Review of Radicals When the index of the radical is not shown then it is understood to be an index of 2. Radical
MULTIPLICATION PROPERTY of RADICALS What do you notice?
MULTIPLICATION PROPERTY of RADICALS where n is a natural number, and a and b are real numbers
Simplify each radical. Write each radical as a product of prime factors, then simplify. Since √80 is a square root. Look for factors that appear twice
Simplify each radical. Write each radical as a product of prime factors, then simplify. Since ∛ 144 is a cube root. Look for factors that appear three times
Simplify each radical. Write each radical as a product of prime factors, then simplify. Since ∜ 162 is a fourth root. Look for factors that appear four times
POWERPOINT PRACTICE PROBLEM Simplify each radical.
Some numbers such as 200 have more than one perfect square factor: For example, the factors of 200 are: 1, 2,4, 5, 8, 10, 20, 25, 40, 50, 100, 200 Since 1, 4, 16, 25, 100, and 400 are perfect squares, we can simplify √400 in several ways: Writing Radicals in Simplest Form
10√2 is in simplest form because the radical contains no perfect square factors other than 1
Mixed Radical: the product of a number and a radical Entire Radical: the product of one and a radical
Writing Mixed Radicals as Entire Radicals Any number can be written as the square root of its square! Any number can be also written as the cube root of its cube, or the fourth root of its perfect fourth!
Writing Mixed Radicals as Entire Radicals
Write each mixed radical as an entire radical
POWERPOINT PRACTICE PROBLEM Write each mixed radical as an entire radical
HOMEWORK O PAGES: O PROBLEMS: 4, 5, 7, 11 – 12 (a, d, e, h, i), 15 – 18, 19,