Changing Bases

Base 10: example number ³ 10² 10¹ 10 ⁰ ₁₀ 10³∙2 + 10²∙1 + 10¹∙ ⁰ ∙0 = 2120 ₁₀ Implied base 10 Base 8: 4110 ₈ 8³ 8² 8¹ 8 ⁰ ₈ 8³∙4 + 8²∙1 + 8¹∙1 + 8 ⁰ ∙0 = 2120 ₁₀ Base 8

Problem Solving: 3, 2, 1, … lets go!

Express the base 4 number 321 ₄ as a base ten number.

Answer: 57

Add: 23 ₄ + 54 ₈ = _______ ₁₀ (Base 10 number)

Answer: 55

Subtract: ₄ ₆ = ______ ₁₀ (Base 10 number)

Answer: 15 ⁴ ³⁄ ₄₈

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

Answer: ₂

Add: ₁₀ ₁₀ = ________ ₁₀ (Base 10 number)

Answer:

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: = 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 Review

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.

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

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

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

EXAMPLE 3: Find the square of each radical expression. Squaring Radical Expressions 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 SquaresRational Square Roots 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: Tell whether each square root is rational, irrational, or not a real number. Identifying Types of Square Roots Solution: Not all irrational numbers are square roots of integers. For example  (approx ) is a irrational number that is not an square root of an integer.

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 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, then a 2 + b 2 = c 2. Note: The hypotenuse, c, is always the longest side.

Find the length of the hypotenuse if 1. a = 12 and b = = c = c = c 2 Take the square root of both sides. 20 = c

= c = c 2 74 = c 2 Take the square root of both sides = c Find the length of the hypotenuse if 2. a = 5 and b = 7.

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

Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = b 2 = b 2 = 100 Solve for b b 2 = b 2 = 84 b = 9.17

Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7. a = 10 2 a = 100 Solve for a. a 2 = a 2 = 51 a = 7.14

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

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 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 = ( ) 2

Determine whether the triangle is a right triangle given the sides 6, 9, and 1.Yes 2.No 3.Purple

EXAMPLE 6 Using the Pythagorean Formula Find the length of the unknown side in each right triangle ? 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:

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

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

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

EXAMPLE 10 Find each root. Finding Other Roots 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

9.2.1: 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 Using the Product Rule to Multiply Radicals 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. Using the Product Rule to Simplify Radicals Solution:

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

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. Solution: Using the Quotient Rule to Simply Radicals

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

EXAMPLE 6 Simplify. Using Both the Product and Quotient Rules 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 Simplify each radical. Assume that all variables represent positive real numbers. Simplifying Radicals Involving Variables 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,

EXAMPLE 8 Simplify each radical. Simplifying Other Roots 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. Simplifying Cube Roots Involving Variables Solution:

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 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

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.

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

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

Rationalizing the Denominator 1. Rationalize denominators with square roots. 2. Write radicals in simplified form. 3. Rationalize denominators with cube roots

9.4.1: Rationalize denominators with square roots. It is easier to work with a radical expression if the denominators do not contain any radicals. This process of changing the denominator from a radical, or irrational number, to a rational number is called rationalizing the denominator. The value of the radical expression is not changed; only the form is changed, because the expression has been multiplied by 1 in the form of

EXAMPLE 1 Rationalize each denominator. Solution: Rationalizing Denominators

9.4.2: Write radicals in simplified form. A radical is considered to be in simplified form if the following three conditions are met. 1. The radicand contains no factor (except 1) that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on. 2. The radicand has no fractions. 3. No denominator contains a radical.

EXAMPLE 2 Solution: Simplifying a Radical

EXAMPLE 3 Simplify Simplifying a Product of Radicals Solution:

EXAMPLE 4 Simplify. Assume that p and q are positive numbers. Simplifying a Quotient of Radicals Solution:

EXAMPLE 5 Simplify. Assume that r and t represent nonnegative real numbers. Simplifying a Radical Quotient Solution:

9.4.3: Rationalize denominators with cube roots.

EXAMPLE 6 Rationalize each denominator. Rationalizing Denominators with Cube Roots Solution:

More Simplifying and Operations with Radicals 1. Simplify products of radical expressions. 2. Use conjugates to rationalize denominators of radical expressions. 3. Write radical expressions with quotients in lowest terms

More Simplifying and Operations with Radicals The conditions for which a radical is in simplest form were listed in the previous section. A set of guidelines to use when you are simplifying radical expressions follows:

More Simplifying and Operations with Radicals (cont’d)

9.5.1: Simplify products of radical expressions.

EXAMPLE 1A Find each product and simplify. Solution: Multiplying Radical Expressions (cont’d)

EXAMPLE 1B Find each product and simplify. Solution: Multiplying Radical Expressions

EXAMPLE 2 Find each product. Assume that x ≥ 0. Solution: Using Special Products with Radicals Remember only like radicals can be combined!

Using a Special Product with Radicals. Example 3 uses the rule for the product of the sum and difference of two terms,

EXAMPLE 3 Find each product. Assume that Using a Special Product with Radicals Solution:

9.5.2: Use conjugates to rationalize denominators of radical expressions. The results in the previous example do not contain radicals. The pairs being multiplied are called conjugates of each other. Conjugates can be used to rationalize the denominators in more complicated quotients, such as To simplify a radical expression, with two terms in the denominator, where at least one of the terms is a square root radical, multiply numerator and denominator by the conjugate of the denominator.

EXAMPLE 4A Simplify by rationalizing each denominator. Using Conjugates to Rationalize Denominators Solution:

EXAMPLE 4B Simplify by rationalizing each denominator. Assume that Using Conjugates to Rationalize Denominators (cont’d) Solution:

9.5.3: Write radical expressions with quotients in lowest terms.

EXAMPLE 5 Write in lowest terms. Solution: Writing a Radical Quotient in Lowest Terms

Solving Equations with Radicals 1. Solve radical equations having square root radicals. 2. Identify equations with no solutions. 3. Solve equations by squaring a binomial. 4. Solve radical equations having cube root radicals

Solving Equations with Radicals. A radical equation is an equation having a variable in the radicand, such as or

To solve radical equations having square root radicals, we need a new property, called the squaring property of equality. Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation : Solve radical equations having square root radicals. If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.

EXAMPLE 1 Solve. Solution: Using the Squaring Property of Equality It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.

EXAMPLE 2 Solve. Using the Squaring Property with a Radical on Each Side Solution:

9.6.2: Identify equations with no solutions.

EXAMPLE 3 Solve. Solution: Using the Squaring Property when One Side Is Negative False Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution. Check:

Solving a Radical Equation. Use the following steps when solving an equation with radicals. Step 1Isolate a radical. Arrange the terms so that a radical is isolated on one side of the equation. Step 6Check all proposed solutions in the original equation. Step 5Solve the equation. Find all proposed solutions. Step 4Repeat Steps 1-3 if there is still a term with a radical. Step 3Combine like terms. Step 2Square both sides.

EXAMPLE 4 Solve Solution: Using the Squaring Property with a Quadratic Expression Since x must be a positive number the solution set is Ø.

9.6.3: Solve equations by squaring a binomial.

EXAMPLE 5 Solve Solution: Using the Squaring Property when One Side Has Two Terms Since x must be positive the solution set is {4}. or

EXAMPLE 6 Solve. Rewriting an Equation before using the Squaring Property Solution: The solution set is {4,9}. or

Solve equations by squaring a binomial. Errors often occur when both sides of an equation are squared. For instance, when both sides of are squared, the entire binomial 2 x + 1 must be squared to get 4 x x + 1. It is incorrect to square the 2 x and the 1 separately to get 4 x

EXAMPLE 7 Solve. Using the Squaring Property Twice Solution: The solution set is {8}.

9.6.4: Solve radical equations having cube root radicals.

Solve radical equations having cube root radicals. We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.

EXAMPLE 8 Solve each equation. Solving Equations with Cube Root Radicals Solution: or

Rational Exponents Fraction Exponents

Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.

How would we simplify this expression? What does the fraction exponent do to the number? The number can be written as a Radical expression, with an index of the denominator.

The Rule for Rational Exponents

Write in Radical form

Write each Radical using Rational Exponents

What about Negative exponents Negative exponents make inverses.

What if the numerator is not 1 Evaluate

What if the numerator is not 1 Evaluate

For any nonzero real number b, and integer m and n Make sure the Radical express is real, no b<0 when n is even.

Simplify

Competition Problems

Which number does not belong in the set? A. B. C. D.

Solve

Answer: 100/9

Simplify

Answer:

Solve for

Answer: 3