Unit 6: Square Roots, Factorials and Permutations Section 1: Introduction to Square Roots If A = s², then s is the square root of A You need to know at least the first ten perfect squares and their square roots Ex1. Name the first 10 perfect squares All non-negative numbers have real square roots, but many of them are irrational Numbers have 2 square roots, a positive and a negative Only give the positive unless you are asked for all

Ex2. Give all square roots of Ex3. If the area of a square is m², what is the length of each side? Radical signs work like a grouping symbol, you are to complete anything within them before taking the square root Ex4. Solve. Round to the nearest hundredth Ex5. Estimate between which two whole numbers the following square root will lie

You can multiply numbers that are under square roots together If you multiply a nonnegative square root by itself, you are left with the integer within the square root i.e. Ex6. Solve and round to the nearest tenth If you are asked to find an exact answer and the square root would be irrational, leave the answer in radical form

Ex7. Give the exact answer to Ex6. This skill is imperative for further math classes Ex8. Solve n² = 361 Ex9. Solve (x + 5)² = 36 Whenever a variable is squared, there will be TWO answers Sections of the book to read: 1-6 and 9-7

Section 2: Simplifying Square Roots To extend the practice of leaving answers in exact form, you can (and must from here on out) simplify radicals and leave them in exact form (unless specified otherwise) The Product of Square Roots Property states that you can multiply square roots together, but you can also use this property in reverse Ex1. Multiply Ex2. Simplify and then multiply

In Ex2. you simplified first, but if you have multiplied first and then simplified, that is known as simplifying the radical Determine the largest perfect square that divides evenly into the number in the question Factor that number out and then find its square root Leave the other number under the radical Ex3. Simplify each radical a) b)c)

Ex4. Simplify Ex5. Multiply and then simplify Simplify each radical expression Ex6. Ex7. Ex8. Section of the book to read: 9-7

Section 3: The Multiplication Counting Principle The Multiplication Counting Principle is used to determine how many possibilities you have when making multiple choices (one after the other) You can only use this principle when your first choice has no affect on the following choices Open your book to page 119, we will be looking at the stadium pictured at the top Ex1. In how many ways can you enter through a south gate and then exit through a north gate?

You can use the Multiplication Counting Principle for more than just two choices Ex2. Mr. Halladay writes a test for his history class. Each of the first 5 questions is multiple choice with 4 choices. The last 6 questions are true-false questions. How many ways can a person answer these questions? Ex3. What is the probability that someone who is randomly guessing on Mr. Halladay’s test will get all questions correct (as a fraction)?

Ex4. A license plate has 4 letters followed by 3 numbers. How many license plates are possible? Ex5. What if you could not use the letters I and O in Ex4. How many license plates are now possible? You can use tree diagrams to visually display possibilities (see page 120) Ex6. A test has 10 questions, each with x number of choices and it has y true-false questions, how many possible ways can you answer this test? Section of the book to read: 2-9

Section 4: Factorials and Permutations Factorials are used when you are making choices, similar to the Multiplication Counting Principle, but now you CANNOT reuse any options The symbol for factorial is ! 5! = 5 · 4 · 3 · 2 · 1 Factorials multiply each number (starting with the given number), decreasing by 1 at a time, until you get to the number 1 Most calculators have a factorial button (locate yours now)

When you cannot reuse options, each way of ordering elements is called a permutation Ex1. A soccer coach has to write down the name of all 11 of her players. In how many ways can she do this? Ex2. Evaluate Sometimes numbers are too large for calculators to evaluate and you have to use the definition of factorial to evaluate Ex3. Evaluate Section of the book to read: 2-10

Section 5: Exponents and Radicals Exponents are not always integers, they can be rational numbers (fractions and decimals) These rational exponents have radical equivalents i.e. All square roots are values raised to the.5 power Square roots are not the only type of roots Cube roots look like this: A cube root is asking you to find what number to the third power = the number under the radical symbol

Ex1. Find Calculators do have a button for other roots Ex2. Find Higher power roots also exist, but they do not have special names Find each root Ex3. Ex4.Ex5. Write each root with a rational exponent Ex6.Ex7. Write each term as a root Ex8.Ex9.

Section 6: Rational Exponents Rational exponents do not always have a numerator of 1 With rational exponents, the numerator is the exponents and the denominator is the root i.e. If you are solving these with numbers, you can either find the root first or the exponent first (it does not matter) Ex1. Write with a fractional exponent

Leave the fractions improper (where appropriate) not as mixed numbers Ex2. Write as a radical expression: Solve each question. Show steps. Ex3.Ex4. Ex5.Ex6. Since it doesn’t matter which is done first (exponent or radical), this equivalence is true: