Radicals (a.k.a. –square roots)

What is a square root? This is called a radical. It is the symbol of the square root operation.

} What is a square root? 25 ft2 The mathematical definition… it is the measurement of a side of a square with a known area… } The area of this square is 25ft2. What is the measurement of the side? 25 ft2 How do we find area? Regular definition… the inverse of squaring a number… 52 is 25, so the is 5!

Common Squares and their Square roots Squares of #s Perfect Square Radicals (square roots) 12 1 √1 =1 22 4 √4 = 2 32 9 √9 = 3 42 16 √16 = 4 52 25 √25 = 5 62 36 √36 = 6 72 49 √49 = 7 82 64 √64 = 8 92 81 √81 = 9 102 100 √100 = 10 112 121 √121 = 11 122 144 √144 = 12

How do I simplify… √48 1st List all factors 48 – 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 2nd Identify the largest perfect square It is 16 3rd Write a radical factor tree to simplify

Use the Radical Factor Tree! What does it simplify to? This is the simplest form. It cannot be simplified any further. The answer is

Try this one!! 1st List the factors of 80 Find the largest perfect square What is your final answer?

The correct answer…

Estimating Radicals To estimate a radical: 1st find 2 perfect square radicals that would fall between 2nd Write a decimal estimate that is closest to the value Example: Estimate √90 1st √90 would fall between √81 and √100 2nd √90 is 9 units from 81 and 10 units from 100, so the estimated value is ≈ 9.4

Ex #4 30 √5 = 5 ** The √5 cancel out*** 6 √5 Using your notes, Estimate these Simplify these √10 √10 √50 √50 √75 √75 √56 √56 Part II: Add/Subtract Multiply/Divide Radicals Ex# 1 √3∙ √2 = √6 Ex #2 5√2 + √2 = 6√2 Ex #3 3√5 – 2√5 = √5 Ex #4 30 √5 = 5 ** The √5 cancel out*** 6 √5

1. √5∙ √2 2. 12√2 + √2 3. 30√5 – 20√5 4. 24 √2 6 √2 Try These: Solutions Estimate these Simplify these √10 ≈ 3.2 √10 = √10 √50 ≈ 7.1 √50 = 5√2 √75 ≈ 8.7 √75 = 5√3 √56 ≈ 7.5 √56 = 2 √14 Try These: 1. √5∙ √2 2. 12√2 + √2 3. 30√5 – 20√5 4. 24 √2 6 √2

Part III. Pythagorean Theorem a2 + b2 = c2 a = leg (side of a right triangle) b = leg (side of a right triangle) c = hypotenuse (always across from the right angle)

Lengths of a right triangle To determine if given lengths can be used to create a right triangle, substitute them into the Pythagorean Theorem. **** Remember – the longest length is always the hypotenuse, c. Ex. Can the length 4, 5, 6 be used to create a right triangle? a = 4, b = 5 c = 6 a2 + b2 = c2 42 + 52 = 62 16 + 25 ≠ 36 41 ≠ 36 So, these lengths CANNOT be used to create a right triangle

Lengths of a right triangle To determine if given lengths can be used to create a right triangle, substitute them into the Pythagorean Theorem Ex#2 Can the length 6, 8, 10 be used to create a right triangle? **** Remember – the longest length is always the hypotenuse, c. a = 6, b = 8 c = 10 a2 + b2 = c2 62 + 82 = 102 36 + 64 = 100 100 = 100 So, these lengths CAN be used to create a right triangle