Changing Bases

Base 10: example number 2120

10³ 10² 10¹ 10⁰

2 1 2 0 ₁₀

10³∙2 + 10²∙1 + 10¹∙2 + 10 ∙0 = 2120⁰ ₁₀

Implied base 10

Base 8: 4110₈

8³ 8² 8¹ 8⁰

4 1 1 0 ₈

8³∙4 + 8²∙1 + 8¹∙1 + 8 ∙0 = 2120⁰ ₁₀

Base 8

Hexadecimal Numbers

Hexadecimal numbers are interesting. There are 16 of them!

They look the same as the decimal numbers up to 9, but then there are the letters ("A',"B","C","D","E","F") in place of the decimal numbers 10 to 15. So a single Hexadecimal digit can show 16 different values instead of the normal 10 like this:Decimal: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Hexadecimal: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Problem Solving:

3, 2, 1, … lets go!

Express the base 4 number 321₄ as a base ten number.

Answer:

57

Add:

23₄ + 54₈ = _______₁₀ (Base 10 number)

Answer:

55

Subtract:

123.11₄ - 15.23₆ = ______₁₀

(Base 10 number)

Answer:

15 ⁴³⁄₄₈

Express the base 10 number 493 as a base two number.

Answer:

111101101₂

Add:

347.213₁₀ + 11.428₁₀ =

________₁₀(Base 10 number)

Answer:

358.641

Add:

234 + 324 =

________4(Base 4 number)

Answer:

1214

Add:

234 + 324 =

________10(Base 10 number)

Answer:

1214

Factorials

Factorial symbol ! is a shorthand notation for a special type of

multiplication.

N! is written asN∙(N-1)∙(N-2)∙(N-3)∙ ….. ∙1

Note: 0! = 1

Example: 5! = 5∙4∙3∙2∙1 = 120

Problem Solving:

3, 2, 1, … lets go!

Solve:

6! = _____

Answer:

720

Solve:

5! 3!

Answer:

20

Solve:

5! 3!2!

Answer:

10

Squares

Positive Exponents

“Squared”: a² = a·a

example: 3² = 3·3 = 9

0²=0 6²=36 12²=1441²=1 7²=49 13²=1692²=4 8²=64 15²=2253²=9 9²=81 16²=2564²=16 10²=100 20²=4005²=25 11²=121 25²=625

What is the sum of the first 9 perfect squares?

Answer:

1+4+9+16+25+36+49+64+81=

285

Shortcut:Use this formula

n(n+1)(2n+1)6

Shortcut:Use this formula9(9+1)(2∙9+1)

6

Answer: 285

Square Roots

Evaluating Roots

1. Find square roots.2. Decide whether a given root is rational, irrational, or not a real number.

3. Find decimal approximations for irrational square roots.4. Use the Pythagorean formula.5. Use the distance formula.6. Find cube, fourth, and other roots.

9.19.19.19.1

9.1.1: Find square roots.

•When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a

2.

The positive or principal square root of a number is written with

the symbol .

Find square roots. (cont’d)

0 0

a

Radical Sign Radicand

The symbol , is called a radical sign, always represents the

positive square root (except that ). The number inside the

radical sign is called the radicand, and the entire expression—radical

sign and radicand—is called a radical.

The symbol – is used for the negative square root of a number.

Find square roots. (cont’d)

The statement is incorrect. It says, in part, that a positive number equals a negative number.

9 3

EXAMPLE 1

• Find all square roots of 64.

Solution:

Finding All Square Roots of a Number

Positive Square Root

Negative Square Root

64 8

64 8

EXAMPLE 2:

•Find each square root.Solution:

Finding Square Roots

169

225

13

15

25

64

25

64 5

8

EXAMPLE 3:

•Find the square of each radical expression.

Squaring Radical Expressions

Solution:

17 2

17 17

29 2

29 29

22 3x 222 3x 22 3x

9.1.2: Deciding whether a given root is rational, irrational, or not a real number.

All numbers with square roots that are rational are called perfect squares.

Perfect Squares Rational Square Roots

25

144

4

9

25 5

144 12

4 2

9 3

A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational.

Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number.

-36

EXAMPLE 4:

•Tell whether each square root is rational, irrational, or not a real number.

Identifying Types of Square Roots

27 irrational

36 26 rational

27 not a real number

Solution:

Not all irrational numbers are square roots of integers. For example (approx. 3.14159) is a irrational number that is not an square root of an integer.

9.1.3: Find decimal approximations for irrational

square roots.A calculator can be used to find a decimal approximation even if a number is irrational.

Estimating can also be used to find a decimal approximation for irrational square roots.

EXAMPLE 5: Approximating Irrational Square Roots

Find a decimal approximation for each square root. Round answers to the nearest thousandth.

Solution:

190 13.784048 13.784

99 9.9498743 9.950

Many applications of square roots require the use of the Pythagorean formula.

If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then

9.1.4: Use the Pythagorean formula.

2 2 2.a b c

Be careful not to make the common mistake thinking that

equals .

2 2a b

a b

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

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, thena2 + b2 = c2.

Note: The hypotenuse, c, is always the longest side.

The Pythagorean Theorem

“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”

aa22 + b + b22 = c = c22

Proof

Find the length of the hypotenuse if1. a = 12 and b = 16.

122 + 162 = c2

144 + 256 = c2

400 = c2

Take the square root of both sides.

20 = c

2400 c

52 + 72 = c2

25 + 49 = c2

74 = c2

Take the square root of both sides.

8.60 = c

Find the length of the hypotenuse if2. a = 5 and b = 7.

274 c

Find the length of the hypotenuse given a = 6 and b = 12

1. 1802. 3243. 13.424. 18

Find the length of the leg, to the nearest hundredth, if

3. a = 4 and c = 10.42 + b2 = 102

16 + b2 = 100Solve for b.

16 - 16 + b2 = 100 - 16b2 = 84

b = 9.17

2 84b

Find the length of the leg, to the nearest hundredth, if4. c = 10 and b = 7.

a2 + 72 = 102

a2 + 49 = 100Solve for a.

a2 = 100 - 49a2 = 51

a = 7.14

2 51a

Find the length of the missing side given a = 4 and c = 5

1. 12. 33. 6.44. 9

5. 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

9 + 64 = 7373 = 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 832 + 82 = ( ) 2

7373

Determine whether the triangle is a right triangle given the sides 6, 9, and 45

1. Yes2. No3. Purple

EXAMPLE 6

2 2 213 15a 2 169 225a 2 56a

Using the Pythagorean Formula

7, 24a b

Find the length of the unknown side in each right triangle.

Give any decimal approximations to the nearest thousandth.

15, 13c b

118

?

2 2 27 24 c 249 576 c 2625 c

625c 25

56a 7.483

2 2 28 11b 264 121b 2 57b

57b 7.550

Solution:

EXAMPLE 7 Using the Pythagorean Formula to Solve an Application

A rectangle has dimensions of 5 ft by 12 ft. Find the length

of its diagonal.

5 ft

12 ft

Solution:

2 2 25 12 c 225 144 c

2169 c

169c

13ftc

9.1.5: Use the distance formula.

2 2,x yThe distance between the points and is

1 1,x y

2 2

2 1 2 1 .d x x y y

EXAMPLE 8

• Find the distance between and .

Using the Distance Formula

6,3 2, 4

2 22 6 4 3d

Solution:

224 7d

65d

16 49d

9.1.6: Find cube, fourth, and other roots. • Finding the square root of a number is the inverse of

squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number.

• The nth root of a is written .n a

n a

n a

Radical signIndex

Radicand

In , the number n is the index or order of the radical.

It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.

EXAMPLE 9

• Find each cube root.• Solution:

Finding Cube Roots

3 64

3 27

3 512

4

3

EXAMPLE 10

• Find each root.

Finding Other Roots

4 81

4 81

4 81

5 243

5 243

3

3

Not a real number.

3

3

Solution:

Evaluating Roots

1. Multiply square root radicals.2. Simplify radicals by using the product rule.

3. Simplify radicals by using the quotient rule.

4. Simplify radicals involving variables.5. Simplify other roots.

9.29.29.29.2

9.2.1: Multiply square root radicals.

•For nonnegative real numbers a and b, and

•That is, the product of two square roots is the square root of the product, and the square root of a product is the product

of the square roots.

a b a b .a b a b

It is important to note that the radicands not be negative numbers in the product rule. Also, in general, .x y x y

EXAMPLE 1•Find each product. Assume that

6 11

13 x

Using the Product Rule to Multiply Radicals

Solution:

0.x

3 5

6 11

13 x

10 10

3 5

10 10

15

66

13x

100 10

9.2.2: Simplify radicals using the product rule.

• A square root radical is simplified when no perfect square factor remains under the radical sign.

• This can be accomplished by using the product rule:

a b a b

EXAMPLE 2•Simplify each radical.

500

17

Using the Product Rule to Simplify Radicals

Solution:

60 4 15

100 5

It cannot be simplified further.

2 15

10 5

EXAMPLE 3•Find each product and simplify.

6 2

100 5

Multiplying and Simplifying Radicals

Solution:

10 50

6 2

10 50 500 10 5

12 2 3

9.2.3: Simplify radicals by using the quotient rule.

• The quotient rule for radicals is similar to the product• rule.

EXAMPLE 4•Simplify each radical.

48

3

5

36

Solution:

Using the Quotient Rule to Simply Radicals

4

494

49

2

7

48

3 16 4

5

36

5

6

EXAMPLE 5

• Simplify. Solution:

Using the Quotient Rule to Divide Radicals

8 50

4 5

8 50

4 5

502

5 2 10

2 10

EXAMPLE 6

• Simplify.

Using Both the Product and Quotient Rules

Solution:

3 7

8 2

3 7

8 2

21

16

21

16

21

4

9.2.4: Simplify radicals involving variables.

•Radicals can also involve variables.

•The square root of a squared number is always nonnegative. The absolute value is used to express this.

•The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers

2For any real number , .a a a

, .0x x x

EXAMPLE 7

•Simplify each radical. Assume that all variables represent positive real numbers.

Simplifying Radicals Involving Variables

Solution:6x

8100 p

4

7

y

3x 23 6Since x x

8100 p 410 p

4

7

y

2

7

y

9.2.5: Simplify other roots.• To simplify cube roots, look for factors that are perfect

cubes. A perfect cube is a number with a rational cube root.

• For example, , and because 4 is a rational number, 64 is a perfect cube.

• For all real number for which the indicated roots exist,

3 64 4

n a . 0ndn

n n n

n

a aa b ab b

bb

EXAMPLE 8

•Simplify each radical.

Simplifying Other Roots

Solution:3 108

4 160

416

625

33 27 4 33 4

4 16 10 4 416 10 42 10

4

4

16

625

2

5

Simplify other roots. (cont’d)

• Other roots of radicals involving variables can also be simplified. To simplify cube roots with variables, use the fact that for any real number a,

• This is true whether a is positive or negative.

3 3 .a a

EXAMPLE 9

•Simplify each radical.

Simplifying Cube Roots Involving Variables

Solution:

3 9z

3 68x

3 554t

15

3a

64

3z

22x3 63 8 x

3 3 227 2t t 3 33 227 2t t 3 23 2t t

3 15

3 64

a

5

4

a

Adding and Subtracting Radicals

1. Add and subtract radicals.2. Simplify radical sums and differences.3. Simplify more complicated radical expressions.

9.39.39.39.3

9.3.1: Add and subtract radicals.•We add or subtract radicals by using the distributive

property. For example,8 3 36

8 6 3 .14 3

and 52 2 3,

32 3as well as and 2 3 .

Radicands are different

Indexes are different

Only like radicals—those which are multiples of the same root of the same number—can be combined this way. The preceding example shows like radicals. By contrast, examples of unlike radicals are

Note that cannot be simplified.35 + 5

EXAMPLE 1

• Add or subtract, as indicated.

Solution:

Adding and Subtracting Like Radicals

8 5 2 5 3 11 12 11 7 10

8 2 5

10 5

3 12 11

9 11

It cannot be added by the distributive property.

9.3.2: Simplify radical sums and differences.

• Sometimes, one or more radical expressions in a sum or difference must be simplified. Then, any like radicals that result can be added or subtracted.

EXAMPLE 2

•Add or subtract, as indicated.

Solution:

Adding and Subtracting Radicals That Must Be Simplified

27 12 5 200 6 18332 54 4 2

3 3 2 3

5 3

5 100 2 6 9 2

5 100 2 6 9 2

50 2 18 2

32 2

3 332 27 2 4 2

3 32 3 2 4 2

3 36 2 4 2

310 2

9.3.3: Simplify more complicated radical expressions.

• When simplifying more complicated radical expressions, recall the rules for order of operations.

A sum or difference of radicals can be simplified only if the radicals are like radicals. Thus, cannot be simplified further.

5 3 5 4 5, but 5 5 3

EXAMPLE 3A•Simplify each radical expression. Assume that all variables represent nonnegative real numbers.

Simplifying Radical Expressions

7 21 2 27

7 21 2 27

147 2 27

49 3 2 27

49 3 2 27

7 3 2 27

7 3 2 3 3

7 3 6 3

13 3

6 3 8r r

6 2 2r r

6 3 2 2r r

18 2 2r r

9 2 2 2r r

3 2 2 2r r

5 2r

Solution:

EXAMPLE 3B

•Simplify each radical expression. Assume that all variables represent nonnegative real numbers.

Simplifying Radical Expressions (cont’d)

2y 72 18y

29 8 9 2y y

23 8 3 2y y

23 2 2 3 2y y

26 2 3 2y y

6 2 3 2y y

3 2y

3 2y

3 33 3 5 2 3x x x x

3 34 481 5 24x x

3 33 33 327 3 5 8 3x x x x

3 33 3 10 3x x x x

313 3x x

Solution: