Section 11 – 1 Simplifying Radicals• Multiplication Property of Square Roots: For

every number a > 0 and b > 0, ab a b

• You can multiply numbers that are both under the radical and you can separate a number under a radical into two radicals being multiplied by each other

• First 10 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (know these)

• Simplifying a radical is when you rewrite a radical so that all of the perfect squares have been factored out

• The end result is said to be in simplified radical form

• Simplify each expression

• Ex1. 243

• Ex2.

• Ex3.

• Ex4.

728x

12 32

7 5 3 8x x

• Division Property of Square Roots: For every number a > 0 and b > 0, a a

b b

• You use this property just as you would with multiplication

• Fractions are not allowed to have radicals in the denominator

• You must rationalize the denominator

• 1) Simplify the radical(s)

• 2) Multiply the numerator and denominator by the radical remaining in the denominator

• Simplify

• Ex5.

• Ex6.

• Ex7.

575

48

x

x

3

7

3

11

12x

Section 11-2 The Pythagorean Theorem

• The Pythagorean Theorem is applied only to RIGHT triangles

• You can use this theorem to find the length of missing sides

• The two shortest sides of a right triangle are called the legs (they must meet at a 90° angle)

• The longest side is called the hypotenuse (it is directly across from the 90° angle)

• The Pythagorean Theorem: a² + b² = c² where a and b are the legs and c is the hypotenuse

• Find the length of the missing side (to the nearest tenth)

• Ex1. a = 8, b = 12, c = ?

• Ex2. a = 20, b = ?, c = 41

• Ex3.

25 ft67 ft

x

Section 11 – 3 The Distance and Midpoint Formulas

• The Pythagorean Distance Formula: The distance d between any two points (x1, y1) and (x2, y2) is

• Ex1. Find the distance between A(-3, 7) and B(5, -4). Show work.

• Ex2. Find the perimeter of ∆XYZ with X(-3, 4) Y(-1, -5) and Z(2, 2). Show work.

• The Pythagorean Distance formula can be derived from the Pythagorean Theorem

2 2

2 1 2 1d x x y y

• If you are asked to give an answer in exact form, you are to give it in simplified radical form

• The midpoint of a segment is the point that divides the segment into two equal segments

• The Midpoint Formula: The midpoint M of a line segment with endpoints A(x1, y1) and B(x2, y2) is

• Ex3. If segment CD has endpoints C(-4, 7) and D(3, -2), find the midpoint of CD. Show work.

1 2 1 2,2 2

x x y y

Section 11 – 4 Operations with Radical Expressions

• Radicals are like radicals if they have the same radicand (the same number under the radical symbol)

• Unlike radicals have different numbers under the radical

• You can add and subtract like radicals, just as you could with like terms

• Ex1. SimplifyA) B)4 5 7 5 3 5 8 20

• You can distribute with radicals as well (remembering that you can multiply radicals together and then simplify them if possible)

• Ex2. Simplify

• Ex3. Use FOIL & then simplify

• Conjugates are the sum and the difference of the same two terms (i.e. and are conjugates)

• The product of two conjugates results in the difference of two squares (an integer)

10 6 5

3 2 7 3 4 7

3 7 3 7

• To rationalize the denominator of an expression that has an addition or subtraction radical expression in the denominator, you must multiply the numerator and denominator by the conjugate of the denominator

• Ex4. Rationalize the denominator

• You should never leave a radical in the denominator of the a fraction!

5

3 11

Section 11 – 5 Solving Radical Equations

• A radical equation is an equation that has a variable as a radicand

• Remember that the expression under a radical must be nonnegative

• Ex1. Solve each equation.

a) b)

• If an equation has radical expressions on both sides, square each side and then solve

8 6a 7 9x

• Ex2. Solve

• When you solve an equation by squaring each side, you create a new equation. This new equation may have solutions that do not solve the original equation. See page 609

• These solutions that do not solve the original equation are called extraneous solutions

• Ex3. Solve

a) b)

• You should make a table of values to create an accurate graph

3 7 5 13m m

8m m 6 9 4x

Section 11 – 6 Graphing Square Root Functions

• A square root function is a function that contains the independent variable in the radicand

• The parent function for square root functions is

• The graph of the parent function is the positive half (because radicands can’t be negative) of a sideways parabola (see page 614)

y x

• The domain of a function contains all possible values of the independent variable

• The domain of the parent function is {x: x > 0}

• You can find the domain by graphing and looking at the graph or you can determine algebraically what values can meaningfully be substituted for x

• Ex1. Find the domain of

• The equation is a translation of the parent function by k units up

2y x

y x k

• The equation is a translation of the parent function by k units down

• The equation is a translation of the parent function by h units to the left

• The equation is a translation of the parent function by h units to the right

• Ex2. Graph each equation

a) b)

y x k

y x h

y x h

3y x 5y x

Section 11 – 7 Trigonometric Ratios• There are three trigonometric ratios: sine

(sin), cosine (cos), and tangent (tan)

• These ratios describe a specific relationship between an angle in a RIGHT triangle and two of the sides of that triangle

• SOHCAHTOA should help you remember these ratios (if you spell it correctly)

cosadjacent

hypotenusesin

opposite

hypotenuse tan

opposite

adjacent

• Use a capital letter to represent an angle

• Open to page 621 to see how to identify adjacent leg vs. opposite leg vs. hypotenuse

• Ex1. Use the triangle below to find

a) sin X b) cos X c) tan X

3 ft

4 ft

5 ft

X

Y Z

• You can use your calculator to find the value of trigonometric functions

• Make sure your calculator mode is in degrees!

• Ex2. Find the value of each expression. Round to the nearest thousandth.

a) sin 130° b) cos 130° c) tan 130°

• You can use SOHCAHTOA to find the lengths of missing sides of a right triangle

• Ex3. Find the length of x. 37°

x29

• An angle of elevation is an angle from the horizontal up to a line of sight (see page 623)

• An angle of depression is an angle measured below the horizontal line of sight (see page 624)

• You can use angle of elevation and angle of depression with trigonometric functions to solve for missing lengths (see example 4 on page 623 and example 5 on page 624)