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Unit 4: Quadratics Revisited. LG 4-1: Rational Exponents LG 4-2: The Discriminant Test March 20 th (Don’t forget the EOC is 3/15 and 3/16). LG 4-1 Rational Exponents. Essential Question: How are rational exponents and roots of expressions similar ?
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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)
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 • Nthroots 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
1. Multiplying terms with exponents: Keep base and ADD exponents Review Exponent Rules 2. Raising an exponent to an exponent: MULTIPLY exponents 3. Division: SUBTRACT exponents with like bases – leftovers go where the higher exponent was
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.
Practice • Review the worked out problems. • Complete as many as possible before we come back together for our next topic.
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.
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!
To Multiply or Divide Radicals are easiest when they have the same ROOT.Outside times outsideRadicand times Radicand
Rewriting Radicals to Rational Exponents Power is on top Roots are in the ground
SIMPLIFY 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
