Find square roots. Find cube roots. 7.1 Objective The student will be able to:

If x 2 = y then x is a square root of y. In the expression is the radical sign and 64 is the radicand. 1. Find the square root: 8 2. Find the square root: -0.2

11, Find the square root: 21 5.Find the square root: -5/9 3. Find the square root:

The index of a cube root is always 3. The cube root of 64 is written as. Cube Roots

What does cube root mean? The cube root of a number is… …the value when multiplied by itself three times gives the original number.

Cube Root Vocabulary index radicand radical sign

If a number is a perfect cube, then you can find its exact cube root. A perfect cube is a number that can be written as the cube (raised to third power) of another number. Perfect Cubes

What are Perfect Cubes? 1 3 = 1 x 1 x 1 = = 2 x 2 x 2 = = 3 x 3 x 3 = = 4 x 4 x 4 = = 5 x 5 x 5 = 125 and so on and on and on…..

Examples: because

Examples :

Not all numbers or expressions have an exact cube root as in the previous examples. If a number is NOT a perfect cube, then you might be able to SIMPLIFY it. Simplify Cube Roots

2 Extract the cube root of the factor that is a perfect cube. 1 Write the radicand as a product of two factors, where one of the factors is a perfect cube. To simplify a cube root... 3 The factors that are not perfect cubes will remain as the radicand.

perfect cube Examples: 1) 2) 3)

Not all cube roots can be simplified! 30 is not a perfect cube. 30 does not have a perfect cube factor. Example: cannot be simplified!

7.2 Objective The student will be able to: use the Pythagorean Theorem

What is a right triangle? It is a triangle which has an angle that is 90 degrees. The two sides that make up the right angle are called legs. The side opposite the right angle is the hypotenuse. leg hypotenuse right angle

The Pythagorean Theorem In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a 2 + b 2 = c 2. Note: The hypotenuse, c, is always the longest side.

Find the length of the hypotenuse if 1. a = 12 and b = = c = c = c 2 Take the square root of both sides. 20 = c

= c = c 2 74 = c 2 Take the square root of both sides = c Find the length of the hypotenuse if 2. a = 5 and b = 7.

3. Find the length of the hypotenuse given a = 6 and b =

Find the length of the leg, to the nearest hundredth, if 4. a = 4 and c = b 2 = b 2 = 100 Solve for b b 2 = b 2 = 84 b = 9.17

Find the length of the leg, to the nearest hundredth, if 5. c = 10 and b = 7. a = 10 2 a = 100 Solve for a. a 2 = a 2 = 51 a = 7.14

6. Find the length of the missing side given a = 4 and c =

7. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle., 3, and 8 Which side is the biggest? The square root of 73 (= 8.5)! This must be the hypotenuse (c). Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

= = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle. Sides:, 3, and = ( ) 2

8. Determine whether the triangle is a right triangle given the sides 6, 9, and 1.Yes 2.No 3.Purple

7.3 Objectives The student will be able to: 1. simplify square roots, and 2.simplify radical expressions.

1 1 = = = = = = 36 49, 64, 81, 100, 121, 144,... What numbers are perfect squares?

1. Simplify Find a perfect square that goes into 147.

2. Simplify Find a perfect square that goes into 605.

Simplify

Look at these examples and try to find the pattern… How do you simplify variables in the radical? What is the answer to ? As a general rule, divide the exponent by two. The remainder stays in the radical.

Find a perfect square that goes into Simplify 5. Simplify

Simplify 1.3x 6 2.3x x 6 4.9x 18

Multiply the radicals. 7. Simplify

8. Simplify Multiply the coefficients and radicals.

9.Simplify

How do you know when a radical problem is done? 1.No radicals can be simplified. Example: 2.There are no fractions in the radical. Example: 3.There are no radicals in the denominator. Example:

10. Simplify. Divide the radicals. Uh oh… There is a radical in the denominator! Whew! It simplified!

11. Simplify Uh oh… Another radical in the denominator! Whew! It simplified again! I hope they all are like this!

12. Simplify Since the fraction doesn’t reduce, split the radical up. Uh oh… There is a fraction in the radical! How do I get rid of the radical in the denominator? Multiply by the “fancy one” to make the denominator a perfect square!

7.4 Objective The student will be able to: simplify radical expressions involving addition and subtraction.

1. Simplify. Just like when adding variables, you can only combine LIKE radicals. Which are like radicals? 2. Simplify.

Simplify

Perimeter = Add all of the sides 3. Find the perimeter of a rectangle whose length is and whose width is

Simplify each radical. 4. Simplify. Combine like radicals.

5. Simplify

Simplify

Simplify