Changing Bases

Base 10: example number 2120 2 1 2 0 ₁₀ Base 8: 4110₈ 4 1 1 0 ₈ 10³ 10² 10¹ 10⁰ 2 1 2 0 ₁₀ 10³∙2 + 10²∙1 + 10¹∙2 + 10⁰∙0 = 2120₁₀ Implied base 10 Base 8: 4110₈ 8³ 8² 8¹ 8⁰ 4 1 1 0 ₈ 8³∙4 + 8²∙1 + 8¹∙1 + 8⁰∙0 = 2120₁₀ 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 15 Hexadecimal: 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

123.11₄ - 15.23₆ = ______₁₀ (Base 10 number) 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 as N∙(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=144 1²=1 7²=49 13²=169 2²=4 8²=64 15²=225 3²=9 9²=81 16²=256 4²=16 10²=100 20²=400 5²=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 formula 9(9+1)(2∙9+1) 6 Answer: 285

Square Roots

9.1 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.

9.1.1: 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.

Find square roots. (cont’d) The positive or principal square root of a number is written with the symbol . The symbol – is used for the negative square root of a number. 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.

Find square roots. (cont’d) The statement is incorrect. It says, in part, that a positive number equals a negative number.

EXAMPLE 1 Find all square roots of 64. Solution: Finding All Square Roots of a Number Find all square roots of 64. Solution:

Finding Square Roots EXAMPLE 2: Find each square root. Solution:

EXAMPLE 3: Find the square of each radical expression. Solution: Squaring Radical Expressions Find the square of each radical expression. Solution:

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 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.

EXAMPLE 4: Identifying Types of Square Roots Tell whether each square root is rational, irrational, or 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:

9.1.4: Use the Pythagorean formula. 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 Be careful not to make the common mistake thinking that equals .

What is a right triangle? hypotenuse leg right angle leg 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.

The Pythagorean Theorem In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a2 + 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.” a2 + b2 = c2

Proof

Find the length of the hypotenuse if 1. a = 12 and b = 16. 122 + 162 = c2 144 + 256 = c2 400 = c2 Take the square root of both sides. 20 = c

Find the length of the hypotenuse if 2. a = 5 and b = 7. 52 + 72 = c2 25 + 49 = c2 74 = c2 Take the square root of both sides. 8.60 = c

Find the length of the hypotenuse given a = 6 and b = 12 180 324 13.42 18

Find the length of the leg, to the. nearest hundredth, if 3 Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = 10. 42 + b2 = 102 16 + b2 = 100 Solve for b. 16 - 16 + b2 = 100 - 16 b2 = 84 b = 9.17

Find the length of the leg, to the nearest hundredth, if 4 Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7. a2 + 72 = 102 a2 + 49 = 100 Solve for a. a2 = 100 - 49 a2 = 51 a = 7.14

Find the length of the missing side given a = 4 and c = 5 1 3 6.4 9

5. The measures of three sides of a triangle are given below 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.

Sides: , 3, and 8 32 + 82 = ( ) 2 9 + 64 = 73 73 = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

Determine whether the triangle is a right triangle given the sides 6, 9, and Yes No Purple

EXAMPLE 6 Find the length of the unknown side in each right triangle. Using the Pythagorean Formula EXAMPLE 6 Find the length of the unknown side in each right triangle. Give any decimal approximations to the nearest thousandth. Solution: 11 8 ?

EXAMPLE 7 A rectangle has dimensions of 5 ft by 12 ft. Find the length 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:

9.1.5: Use the distance formula. The distance between the points and is

EXAMPLE 8 Find the distance between and . Solution: Using the Distance Formula EXAMPLE 8 Find the distance between and . Solution:

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 Radical sign Index 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.

Finding Cube Roots EXAMPLE 9 Find each cube root. Solution:

EXAMPLE 10 Finding Other Roots Find each root. Solution:

9.2 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.

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

EXAMPLE 1 Find each product. Assume that Solution: Using the Product Rule to Multiply Radicals Find each product. Assume that Solution:

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:

EXAMPLE 2 Simplify each radical. Solution: Using the Product Rule to Simplify Radicals Simplify each radical. Solution:

EXAMPLE 3 Solution: Find each product and simplify. Multiplying and Simplifying Radicals Find each product and simplify. Solution:

9.2.3: Simplify radicals by using the quotient rule. The quotient rule for radicals is similar to the product rule.

EXAMPLE 4 Solution: Simplify each radical. Using the Quotient Rule to Simply Radicals Simplify each radical. Solution:

EXAMPLE 5 Simplify. Solution: Using the Quotient Rule to Divide Radicals Simplify. Solution:

EXAMPLE 6 Simplify. Solution: Using Both the Product and Quotient Rules Simplify. Solution:

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

EXAMPLE 7 Simplifying Radicals Involving Variables Simplify each radical. Assume that all variables represent positive real numbers. Solution:

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,

Simplifying Other Roots EXAMPLE 8 Simplify each radical. Solution:

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.

EXAMPLE 9 Simplify each radical. Solution: Simplifying Cube Roots Involving Variables Simplify each radical. Solution:

Adding and Subtracting Radicals 9.3 Adding and Subtracting Radicals 1. Add and subtract radicals. 2. Simplify radical sums and differences. 3. Simplify more complicated radical expressions.

9.3.1: Add and subtract radicals. We add or subtract radicals by using the distributive property. For example, 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.

EXAMPLE 1 Add or subtract, as indicated. Solution: Adding and Subtracting Like Radicals Add or subtract, as indicated. Solution: 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 Add or subtract, as indicated. Solution:

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.

Simplifying Radical Expressions EXAMPLE 3A Simplify each radical expression. Assume that all variables represent nonnegative real numbers. Solution:

EXAMPLE 3B Simplifying Radical Expressions (cont’d) Simplify each radical expression. Assume that all variables represent nonnegative real numbers. Solution: