UNIT 6Radical Functions and Right Triangles

SECTION 1: INTRODUCTION TO SQUARE ROOTS

The square root of a number is the value that when multiplied by itself is equal to the original number

For example, what number multiplied by itself is equal to 4? 2 times 2 = 4, therefore 2 is the square root of 4

Ex1. Find A perfect square is a number that has a

counting number as its’ square root Ex2. Name the first 10 perfect squares

You need to know these! Numbers have two square roots (one positive

and one negative), but you only have to give the positive unless you are asked for both

49

Ex2. Give all square roots of 156.25 Ex3. If the area of a square is 9.2416 m², what

is the length of each side? Use a radical symbol like a grouping symbol Ex4. Solve and round to the nearest hundredth If numbers are both under radical symbols, you

can multiply the numbers together (this is the Product of Square Roots Property) i.e.

If the two numbers under the radical are the same, the solution will just be the number i.e.

If the directions say to find the exact answer, leave the solution under the radical

Ex5. Find the exact answer

2 212 3

5 7 35

5 5 25 5

215 2 6

If a variable is squared, there will be two solutions

Ex6. Solve n² = 361 Ex7. Solve (x + 5)² = 36 Sections of the book to read: 1-6 and 9-7

SECTION 2: SIMPLIFYING SQUARE ROOTS Simplifying a radical means to write it in such a

way that there are NO perfect square factors to the number under the radical You will need to make a factor tree to simplify

We will be using the Product of Square Roots Property in reverse in order to write the answers in simplified radical form

Ex1. Simplify Ex2. Multiply You can choose to multiply and then simplify or

you can simplify and then multiply (you may have to simplify twice if you do it in that order)

Determine the largest perfect square factor in order to only go through the process once

203 5 2 7

Simplify each radical Ex3. Ex4. Ex5. Ex6.

You can also simplify radicals containing variables

If the exponent is even, the exponent is cut in half and placed outside of the parentheses

If the exponent is odd, you will have to factor it into two factors (the second factor will be to the first power)

Ex7. Simplify

Simplify each radical expression Ex8. Ex9. Ex10.

Section of the book to read: 9-7

60 700

2 22 5 14 8 72

2

6 11 14x y z

2 9 1524x y z 4 7 12300a b c 15 175

5

SECTION 3: EXPONENTS AND RADICALS Exponents do not have to be integers, they can

be fractions or decimals These rational exponents have radical

equivalents i.e.

A cube root is asking for the number that multiplied by itself 3 times equals the number in question i.e.

You can type these into your calculator as a root or as a rational exponent

Ex1. Find Other roots do not have special numbers (just

say “fourth root”, “fifth root”, etc.)

1

2x x

13 3x x

3 1331

Find each root Ex2. Ex3. Ex4.

Write each root with a rational exponent Ex5. Ex6.

Write each term as a root Ex7. Ex8.

4 2401 7 279936 101024

5 m 7 ab

1

6c1

8z

SECTION 4: RATIONAL EXPONENTS Rational exponents can have numerators other

than one, creating a radical that also has an exponent

The numerator is the exponent and the denominator is the root i.e.

You can find the root and then the exponent or the exponent and then the root (it doesn’t matter the order)

Ex1. Write with a fractional exponent Leave the fractions improper, do not change to

a mixed number Ex2. Write as a radical expression

2 2

3 2 33x x x

5 8n

7

9v

Solve each question. Ex3. Ex4.

Ex5. Ex6.

5

243 28

54 814

3125

SECTION 5: RADICAL EQUATIONS Any equation with the variable in a radicand

(under the radical) is a rational equation Solve these equation by isolating the radicand,

squaring both sides, and then solving like a regular equation

Check to make sure that your answer is possible You may get no solutions

Solve each equation. Check each solution. Ex1. Ex2.

Ex3. Ex4.

3 4x 3 4x

2 6 4x 3 2 6n n

SECTION 6: THE PYTHAGOREAN THEOREM You can only use the Pythagorean Theorem with

RIGHT triangles The longest side of the triangle is called the

hypotenuse The two other sides are called the legs The Pythagorean Theorem: a² + b² = c²

The legs are a and b, the hypotenuse is c

Some solutions may be rounded, and others may need to be exact

Ex1. Find the missing length to the nearest tenth

Ex2. Find the exact length of the missing side

Ex3. A rectangle has a length of 80 cm and a width of 42 cm. What is the length of the diagonal (nearest hundredth)?

Section of the book to read: 1-8

15 ft

17 ft

x

23 m

9m

y

SECTION 7: SOHCAHTOA Right triangles form trigonometric ratios The three trigonometric ratios are: sine (sin),

cosine (cos), and tangent (tan) Memorize the three ratios:

Ex1. Identify which side is opposite, adjacent, and the hypotenuse from angle A.

Write the answers as simplified ratios (not mixed numbers)

sinopposite

hypotenuse cos

adjacent

hypotenuse tan

opposite

adjacent

A

Side 1

Side 2

Side 3

Ex2. Find each of the trigonometric ratios A) sin X B) cos X C) tan X D) sin Y E) cos Y F) tan Y

8

6

10

X

Y Z

SECTION 8: USING SOHCAHTOA You can use the trigonometric ratios to find the

lengths of missing sides and the measures of angles

First determine which trigonometric ratio is appropriate for each question

Set up the equation and then solve for the missing piece

Make sure your calculator is in DEGREE mode If you must solve for a missing angle, you will

need to use an inverse trigonometric function (we will practice this)

Always show your set up and your answer

Solve for each missing element Ex1. Ex2.

Ex3. Ex4.

12x

27°53

28

n

34

m

73°

a

4637