R ADICAL E XPRESSIONS AND EQUATIONS Chapter 11. INTRODUCTION We will look at various properties that are used to simplify radical expressions. We will

RADICAL EXPRESSIONS AND EQUATIONSChapter 11

INTRODUCTION

We will look at various properties that are used to simplify radical expressions.

We will then apply properties to problems using the Pythagorean Theorem as well as the Distance and Midpoint formulas and solving special triangles.

We will solve radical expression and graph square root functions.

Finally, solving right triangles we obtain definitions for three key trigonometric functions which can then be used to solve real world problems such as problems involving angles of elevation and depression.

SIMPLIFYING RADICALS (11.1)

Radical expressions: Algebraic expressions that contain the radical sign.

We can simplify a radical expression by removing perfect-square factors from the radicand. We can use the Multiplication Property of Square

Roots to help in the simplification of radical expressions.

Multiplication Property of Square RootsFor every number


We can rewrite the radicand as a product of the perfect-square factors times the remaining factors by using the Multiplication Property of Square Roots.

Sample ProblemSimplify .


Simplifying radical expressions that contain variables is also possible. A variable with a non-zero, even exponent is a

perfect square. A variable with an odd exponent (other than 1

and -1) are the product of a perfect square and the variable.

Sample ProblemSimplify .


We can sometimes use the Multiplication Property of Square Roots to write . Sometimes the product of two radicals has a

perfect square.

Sample ProblemSimplify each radical expression.a)


We can use the Division Property of Square Roots to simplify expressions.

When the denominator of the radicand is a perfect square, it is easier to simplify the numerator and denominator separately.

When the denominator is not a perfect square, it maybe easier to divide first and then simplify the radical expression.

Division Property of Square RootsFor every number


Sample Problem Simplify each radical expression.


When a radicand in the denominator of a radical expression is not a perfect square, we may have to rationalize the denominator to simplify the expression.

To rationalize a radical expression we multiply the numerator and the denominator by the same radical expression. The radical expression we choose should make

the denominator a perfect square. Since we are multiplying the numerator and the

denominator by the same radical expression, we are essentially multiplying by one.

SIMPLIFYING RADICALS (11.1)Sample ProblemSimplify by rationalizing the denominator.


The summary below can help you determine whether a radical expression is in the simplest radical form:The radicand has no perfect-square factors

other than 1.The radicand has no fractions.The denominator of a fraction has no

radical.

THE PYTHAGOREAN THEOREM (11.2)

A right triangle is a triangle that has one angle that is 90o degrees.

The Pythagorean Theorem describes the relationship between the lengths of the sides of the right triangle.

hypotenuse, c

leg, a

leg, b


The Pythagorean Theorem can be used to solve for problems involving right triangles and their sides.

The Pythagorean Theorem In any right triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2


Sample Problem What is the length of the hypotenuse of the right triangle at the right?

c

12 cm

9 cm


We can use the Pythagorean Theorem to find the length of one of the legs if we know the lengths of the hypotenuse and the other leg.

Sample ProblemA fire truck parks beside a building such that the base of the ladder is 16 ft. from the building. The fire truck extends its ladder 30 ft. The ladder sits on top of the truck which is 10 ft. above the ground. How high is the top of the ladder from the ground?


Not all converse statements are true, but the converse of the Pythagorean Theorem is always true. Thus, we can use this converse to determine if a

given triangle is right triangle or if it is not a right triangle.

The Converse of the Pythagorean Theorem If a triangle has sides of lengths a, b, and c, and a2 + b2 = c2, then the triangle is a right triangle with hypotenuse of length c.


Sample ProblemDetermine whether the given lengths can be sides of a right triangle.a) 5 in., 12 in., 13 in.b) 7 m, 9 m, 12 m


The Pythagorean Theorem and its converse has many uses in solving problems in physics. One use is in solving for problems involving

forces.Sample ProblemIf two forces pull at right angles to each other, the resultant force is represented as the diagonal of a rectangle. The diagonal forms a right triangle with two of the perpendicular side of the rectangle. For a 30-lb force and a 40-lb force, the resultant force is 50-lbs. Are the forces pulling at right angles to each other?

THE DISTANCE AND MIDPOINT FORMULAS (11.3)

For vertical and horizontal line segments, we can find their lengths by subtracting their y- and x-coordinates, respectively.

For any two points P(x1, y1) and Q(x2, y2) not on a horizontal or vertical line, we can graph the points and form a right triangle. We can then use the Pythagorean Theorem to find the distance between the points. Let’s take a look at how this is done.



We can find an exact value by substituting values into the Distance Formula and simplifying the radical expression.

We can find the approximate distance by using a calculator to estimate when a radical expression is not a perfect square.

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


Sample ProblemFind the distance between T(1,) and V(3, )

THE DISTANCE AND MIDPOINT FORMULAS (11.3) We can use the Distance formula to find the

lengths of the sides of a geometric figure that is drawn on a geometric plane.

Use the Distance formula to find the lengths of the sides then add them together. When adding values that are square roots, use a

calculator to add before you round the answer.A (-2, 2)

D (-3,- 2)

C (3, -3)

B (3,4)

Sample ProblemFind the exact length s of each side of the quadrilateral ABCD. Then find the perimeter to the nearest tenth.


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

Sample Problem:Find the midpoint of the line segment CD with endpoints C(

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

THE DISTANCE AND MIDPOINT FORMULAS (11.3) If we know the coordinates of the endpoints

of a diameter of a circle, we can use the midpoint formula to find the coordinates of the center of the circle.

Sample Problem A circle is drawn on a coordinate plane. The endpoints of a diameter are (-1,5) and (4,-3). What is the center of the circle?

OPERATIONS WITH RADICAL EXPRESSION (11.4) Like radicals have the same radicand and

unlike radicals have different radicands. Example: are like radicals. Like radicals can be combined together using the

Distributive Property. Simplifying radical expressions will allow us

to see if two radicals are the same.

Sample Problem Simplify.

OPERATIONS WITH RADICAL EXPRESSION (11.4) We can also us the Distributive Property to

simplify radical expressions in the form of .


OPERATIONS WITH RADICAL EXPRESSION (11.4)

If both radical expressions have two terms, we can multiply the same way we find the product of two binomials, by using FOIL.


OPERATIONS WITH RADICAL EXPRESSION (11.4) Conjugates are the sum and the difference

of the same two terms. Example:

The product of two conjugates results in a difference of two squares. Example: ()2

= 5 – 2 = 3

The product of these conjugates has no radical. When a denominator contains a radical

rationalize the denominator by the numerator and the denominator by the conjugate of the denominator.

OPERATIONS WITH RADICAL EXPRESSION (11.4)

Sample ProblemSimplify

SOLVING RADICAL EQUATIONS (11.5)

A radical equation is an equation that has a variable in a radicand.

Solving a radical equation requires: 1st: Get the radical by itself on one side of the

equation. 2nd: Square both sides. The radical expression under the radical must

not be negative.


Sample ProblemSolve each equation.


For an equation in the form we can either square both sides first, or we can divide, isolate the radical, then square.

Sample ProblemSolve


For radical equations with radical expressions on both sides of the equal sign, square both sides of the equal sign then solve for the unknown variable.

Sample ProblemSolve


Extraneous solution: A solution that does not satisfy the original equation. When solving an equation by squaring both

sides, we may run the risk of creating extraneous solutions.

Always check all solutions in the original equation to prevent the inclusion of extraneous solutions in the solution set.

Sample ProblemSolve then check the solutions:


In some cases the only solution that is gotten is an extraneous one. In this case, the original equation is determined to have no solution.

Sample ProblemSolve

GRAPHING SQUARE ROOT FUNCTIONS (11.6) A square root function is a function that

contains the independent variable in the radicand. The function is the simplest square root

function. For x-values that are not perfect squares, we can

approximate the y-values to the nearest tenth. We can graph a square root function by

plotting the points. Plot the least value in the domain and several

other points. Connect the points with a smooth curve.

GRAPHING SQUARE ROOT FUNCTIONS (11.6)

Plotting the square root function

x y0 0

1 1

2 1.4

4 2

6 2.4

9 3


For real numbers, the values of the radicand cannot be negative. So the domain is limited to those values of x that

make the radicand greater than or equal to 0. To determine the domain values set the

equation such that the radicand is greater than or equal to 0.

Sample ProblemFind the domain of each function.


For any positive number k, translate the graph of up k units while translates the graph down k units.


For any positive number h, translates the graph of to the left h units, while translates the graph h units to the right.


Other changes to the square root function graph:

TRIGONOMETRIC RATIOS (11.7) Trigonometric ratios: These are ratios of

the lengths of the corresponding sides of similar right triangles and their relations to the corresponding angles.

AC

B

c

b

a

sine A =

cosine A =

tangent A =

TRIGONOMETRIC RATIOS (11.7)

Sample Problem Use the triangle below to find sin A, cos A, and tan A.

b = 12

c = 13a = 5


We can use the calculator to obtain values for the trig functions as well, given the angles. Make sure that your calculator is set on the

“degree” mode.Sample ProblemFind sin 50o.


We can also use the trig. functions to find a missing length of a right triangle given the appropriate angles and lengths of sides.

Sample ProblemFind the value of x in the triangle at the right.

16

x

55o


We can use the trigonometric ratios to measure distances indirectly when we know the angle of elevation or the angle of depression. Angle of elevation: An angle from the

horizontal up to a line of sight. Angle of depression: An angle measured below

the horizontal line of sight.


Sample ProblemSuppose the angle of elevation from a rowboat to the light of a lighthouse is 35o. You know that the lighthouse is 96 ft. tall. How far from the lighthouse is the rowboat.


Sample ProblemA pilot is flying a plane 20,000 ft. above the ground. The pilot begins a 2o descent to an airport runway. How far is the plane from the start of the runway (in ground distance)?

RADICAL EXPRESSIONS AND EQUATIONSChapter 11

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R ADICAL E XPRESSIONS AND EQUATIONS Chapter 11. INTRODUCTION We will look at various properties that are used to simplify radical expressions. We will