Unit 4: Quadratics Revisited LG 4-1: Rational Exponents LG 4-2: The Discriminant Test March 20th (Don’t forget the EOC is 3/15 and 3/16) Day 1

LG 4-1 Rational Exponents Essential Question: How are rational exponents and roots of expressions similar? In this learning goal, we will extend the properties of exponents to rational exponents. In order to do this, sometimes it is beneficial to rewrite expressions involving radicals and rational exponents using the properties of exponents.

ENDURING UNDERSTANDINGS Nth roots are inverses of power functions. Understanding the properties of power functions and how inverses behave explains the properties of nth roots. Real-life situations are rarely modeled accurately using discrete data. It is often necessary to introduce rational exponents to model and make sense of a situation. Computing with rational exponents is no different from computing with integral exponents.

EVIDENCE OF LEARNING By the conclusion of this unit, you should be able to Make connections between radicals and fractional exponents Results of operations performed between numbers from a particular number set does not always belong to the same set. For example, the sum of two irrational numbers (2 + √3) and (2 - √3) is 4, which is a rational number; however, the sum of a rational number 2 and irrational number √3 is an irrational number (2 + √3)

SELECTED TERMS AND SYMBOLS Nth roots: The number that must be multiplied by itself n times to equal a given value. The nth root can be notated with radicals and indices or with rational exponents, i.e. x1/3 means the cube root of x. Rational exponents: For a > 0, and integers m and n, with n > 0, Rational number: A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers. Whole numbers. The numbers 0, 1, 2, 3, ….

The properties of operations: Associative property of addition (a + b) + c = a + (b + c) Commutative property of addition a + b = b + a Additive identity property of 0 a + 0 = 0 + a = a Existence of additive inverses For every a there exists –a so that a + (–a) = (–a) + a = 0. Associative property of multiplication (a × b) × c = a × (b × c) Commutative property of multiplication a × b = b × a Distributive property of multiplication over addition a × (b + c) = a × b + a × c

Review Exponent Rules 1. Multiplying terms with exponents: Keep base and ADD exponents 2. Raising an exponent to an exponent: MULTIPLY exponents 3. Division: SUBTRACT exponents with like bases – leftovers go where the higher exponent was Today should be a review!!!

4. Negative Exponents: MOVE the base to the opposite part of the fraction & it’s no longer negative

Practice simplifying: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Simplify 13.

Practice Review the worked out problems. Complete as many as possible before we come back together for our next topic.

Simplifying & Operating with radicals

No number where the root is means it’s a square root (2)!! Parts of a radical No number where the root is means it’s a square root (2)!!

Simplifying Radicals Break down the radicand in to prime factors. Bring out groups by the number of the root.

Simplify

Simplify

Simplify

Simplify

Operations with Radicals To Add or Subtract Radicals, they must have the SAME Root & SAME Radicand

worrrrrrrrk Please complete the worksheet – you can do as much or as little as you need This does not mean NOTHING! If you know you need help – please help yourself!

Warm UP Day 2

To Multiply or Divide Radicals are easiest when they have the same ROOT. Outside times outside Radicand times Radicand

Rewriting a Radical to have a Rational Exponent

Rewriting Radicals to Rational Exponents Power is on top Roots are in the ground

Rewrite with a Rational Exponent

Rewrite with a Rational Exponent

Rewrite with a Rational Exponent

Rewrite with a Rational Exponent

Rewrite with a Rational Exponent

Rewriting Rational Exponents to Radicals

Rewrite with a Rational Exponent (don’t evaluate)

Rewrite with a Rational Exponent (don’t evaluate)

Rewrite with a Rational Exponent (don’t evaluate)

SIMPLIFY Change to a radical Prime Factor Bring out groups of the root Sometimes you can simplify in the calculator. KNOW how to do it by hand!! Change to a radical Prime Factor Bring out groups of the root

SIMPLIFY

SIMPLIFY

SIMPLIFY

SIMPLIFY

SIMPLIFY

Practice TIME

Warm UP! Simplify: Rewrite: 1) 2) Day 3

The cube below has a volume of 343 cubic inches The cube below has a volume of 343 cubic inches. Find the length of an edge of the cube.

The volume of a basketball is approximately 448. 9 cubic inches The volume of a basketball is approximately 448.9 cubic inches. Find the radius of the basketball to the nearest tenth.

To Multiply or Divide Radicals when they have the different ROOTs, you must rewrite with rational exponents & follow your exponent rules.

5. Simplify:

6. Rewrite and Simplify:

7. Rewrite the rational exponent as a radical and then simplify:

8. Simplify:

9. Simplify:

10. Simplify:

11. Simplify:

12. Simplify:

13. Simplify:

14. Simplify:

Practice

Warm UP! Day 4

Rationalizing the Denominator

that we don’t leave a radical There is an agreement in mathematics that we don’t leave a radical in the denominator of a fraction.

So how do we change the denominator of a fraction? (Without changing the value of the fraction, of course.)

The same way we change the denominator of any fraction! (Without changing the value of the fraction, of course.)

We multiply the denominator and the numerator by the same number.

By what number can we multiply to change it to a rational number?

The answer is . . . . . . by itself!

Remember, is the number we square to get n. So when we square it, we’d better get n.

In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by .

In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by .

Because we are changing the denominator to a rational number, we call this process rationalizing.

Rationalize the denominator: (Don’t forget to simplify)

Rationalize the denominator: (Don’t forget to simplify) (Don’t forget to simplify)

When there is a binomial with a radical in the denominator of a fraction, you find the conjugate and multiply. This gives a rational denominator.

Simplify: Multiply by the conjugate. FOIL numerator and denominator. Next

Simplify · =

Combine like terms Try this on your own:

Warm UP! Don’t‘ forget your registration appointments!!! Simplify the following: 𝑥 1 3 𝑥 5 3 2) 3 𝑥 6 𝑥 2 3 𝑥 8 3) 𝑥 1 2 𝑦 2 3 3 𝑥 1 2 Day 5

MORE PRACTICE… You will practice more of the rules we have learned for the next 25 minutes We will then do a group activity The last 25 minutes of class will be an individual and graded assignment.

Now, clear your desk other than a pencil… and your brain Now, clear your desk other than a pencil… and your brain! NO CALCULATORS! You will now be given a set of challenging problems. The first 4 are for you to work on by yourself. The back are for you to work on with a bud. Remember, you always need to get an answer. TRY SOMETHING. It’s ok if something wrong 

ON YOUR OWN: 1. Simplify 2. Simplify 3. 4.

WITH A PARTNER: 1. 2. Solve for a: 3. Solve for a:

Warm up! Day 6

TODAY You will be in room 1004 with Mr. Osinski. Complete the packet – make sure to READ THE NOTES! Don’t just skip them. You can access the answers and additional notes on my blog. Go to unit 4 and look at slides 99-115. We will go over everything Monday 

QUEEEEZ TIME

The hypotenuse of a right triangle is an irrational number. After the Quiz… DETERMINE IF THE FOLLOWING STATEMENT IS ALWAYS, SOMETIMES, OR NEVER TRUE: The hypotenuse of a right triangle is an irrational number.

The Number Systems There are different subsets of numbers in the Number System Universe… What do you know about the following types of numbers: Reals Imaginary Complex Rationals Irrationals Integers Whole Numbers Natural Numbers

Number Systems Rational Numbers – can be expressed as a ratio of p/q, such that p and q are both integers (but not equal to 0). All rational numbers can be expressed as a terminating or repeating decimal. Irrational Numbers – cannot be expressed as a ratio of p/q and cannot be expressed as a terminating or repeating decimal.

Irrational Numbers… Historically, irrational numbers were difficult to comprehend because they cannot be expressed easily. For example, consider creating a physical representation of 2 . We could accomplish this by creating a right triangle with legs of length 1 meter. The hypotenuse should be the length of 2 meters.

1 meter 1 meter

Irrational Numbers… What is interesting is, no matter how precise a ruler is, you can NEVER measure the exact length of the hypotenuse using a metric scale. The hypotenuse will ALWAYS fall between any two lines of metric division. 2 ≈1.41421356237309504880168872… meters which can NEVER be precisely written as a decimal. It continues forever without repeating a pattern.

Irrational Numbers… This bothered the ancient Greeks – especially Pythagoreans. They thought it was ILLOGICAL or CRAZY that is was possible to draw a line of a length that could NEVER be measured precisely. They even hid the fact that they knew this as they believed it to be an imperfection of mathematics! What’s another word for someone illogical and crazy? IRRATIONAL!

Into which groups does each number go? Natural Whole Integer Rational Irrational Real Imaginary Complex -8 5 3+2i 7 4 9 7i

Into which groups does each number go? Natural Whole Integer Rational Irrational Real Imaginary Complex -8 5 3+2i 7 4 9 7i

Closed Number Systems A set of numbers is said to be closed under an operation if any two numbers from the original set are then combined under the operation and the solution is always IN THE SAME SET as the original numbers. Sound confusing? Here are some examples:

Examples of Closed Number Sets The sum of even numbers always results in an even number. So, the set of even numbers is CLOSED under addition. The sum of odd numbers always results in an even number. So the set of odd numbers is NOT CLOSED under addition. If you can think of ONE instance that doesn’t satisfy the operation, it is not a closed system!

NO! YES! Think about these: Is the set of EVEN NUMBERS closed under DIVISION? Is the set of ODD NUMBERS closed under MULTIPLICATION? NO! Example: 10÷2=5 YES! Example: 5∙3=15

YES! YES! YES! NO! You try: For the set of INTEGERS: Is the set closed under addition? Is the set closed under subtraction? Is the set closed under multiplication? Is the set closed under division? YES! Example: −10+2=-8 YES! Example: 2−10=−8 YES! Example: −10∙2=−20 NO! Example: 5÷2=2.5

YES! YES! NO! NO! You try more! For the set of RATIONAL NUMBERS: Is the set closed under addition? Is the set closed under multiplication? For the set of IRRATIONAL NUMBERS: YES! YES! NO! NO!

Solving Exponential Equations Day 5

WARM UP Are the natural numbers closed under multiplication? Are the natural numbers closed under subtraction? Are the even numbers closed under exponentiation? Are the odd numbers closed under exponentiation? Day 7

IN A GROUP OF 2 OR 3 PEEPS: You are going to make a poster like this one. You’ll see there are three columns, ‘Always True’, ‘Sometimes True’, and ‘Never True’. You will be given some statements to classify on your poster.

LG 4-2: The Discriminant Essential Question: Why is it important to allow solutions for 𝑥 2 +1=0? We will be solving quadratic equations and inequalities in one variable  The methods we will use involve: Solve quadratic equations by inspection (e.g., for x2 = 49) taking square roots Factoring completing the square and the quadratic formula, as appropriate to the initial form of the equation

SELECTED TERMS AND SYMBOLS Binomial Expression: An algebraic expression with two unlike terms. Polynomial function: A polynomial function is defined as a function, f(x)= where the coefficients are real numbers. Standard Form of a Polynomial: To express a polynomial by putting the terms in descending exponent order. Trinomial: An algebraic expression with three unlike terms.

Videos full of help! http://brightstorm.com/search/?k=polynomials http://www.khanacademy.org/search?page_search_query=polynomials

The Quadratic Formula and the Discriminant

*All methods of solving a quadratic equation give the same answer* We have a number of different ways for solving (finding the roots) quadratic equations 1. Graphing 2. Factoring 3. Completing the Square 4. The Quadratic Formula *All methods of solving a quadratic equation give the same answer*

REVIEW: Solve by Taking Roots

REVIEW: Solve by Completing the Square

REVIEW: Solve by using the Quadratic Formula The seventh-century Indian mathematician Brahmagupta was among the first to use a general algebraic formula to find roots of quadratic equations!

The radicand of the quadratic formula. The Discriminant The radicand of the quadratic formula. The Discriminant can be negative, positive or zero If the Discriminant is positive, there are 2 real answers. If the Discriminant is zero, there is 1 real answer. If the Discriminant is negative, there are 2 complex answers. Complex answer have i because they have an imaginary part. If the discriminant is not a perfect square, then there will be 2 irrational roots

Describe the roots Determine the Discriminant and state the type of roots 0, One rational root -11, Two complex roots 80, Two real irrational roots

Describe the roots:

Homework: Describe the roots of the following quadratics:

2 Imaginary Irrational Roots WARM UP Determine the number & types of roots. 2 Imaginary Irrational Roots 1 Real Rational Root 2 Real Rational Roots Day 8

Just the Right Border Task

If x 2 + y 2 = 20 and xy = 15, find the value of (x + y) 2 If x 2 + y 2 = 20 and xy = 15, find the value of (x + y) 2. If (x – y)= 9 and (x + y) = 5, find the value of x 2 – y 2.

Warm UP Day 9

Review Quick Quiz Powers of 2