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Roots and Radical Expressions

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Presentation on theme: "Roots and Radical Expressions"— Presentation transcript:

1 Roots and Radical Expressions
What you’ll learn To find nth roots. Add, subtract, multiply, and divide radical expressions. Vocabulary nth root, principal root, radicand, index

2 Take a note: Corresponding to every power, there is a root.
Example: As there are squares(second powers), there are square roots. As there are cubes (third powers), there are cube roots, and so on. index radicand Type of number b Number of Real nth Roots When n is Odd Roots When n is Even positive 1 2 negative none Radical sign

3 Problem 1: Finding all Real roots
n is 3 and 3 is odd. So there is only one real cube root of a number. Answers: n=4; even. Since 1 is positive, there are two real fourth roots Answers:

4 a)What are the real fifth roots of 0,-1 and 32?
n=4; even. Since is negative, there are no real fourth roots of Answer: No real fourth roots n=4; even. Since 16/81 is positive , there are two real fourth roots. Answer Your turn a)What are the real fifth roots of 0,-1 and 32? b) What ere the square roots of0.01, -1 and ? Answers

5 Problem 2: Finding Real Roots.
There is no real root because there is no real number whose fourth power is -1 It is tempting to conclude that ,but the above problem shows that is not the case If n is even, then is positive even if a itself is negative

6 Your turn Answers: There is no real root because there is no real
number whose fourth power is -81 There is no real root because there is no real number whose square is -49

7 Remember when the radicand contains a variable
Take a note Remember when the radicand contains a variable expression, you MUST include the absolute value when n is even, and omit it when n is odd. Problem 3: Simplifying Radical Expressions The index of the square root is even. However, X to the power 4 is always nonnegative. The index is odd, so you can not include absolute values. The index is even. The absolute value symbols ensure that the root is positive when is negative. Absolute value symbols are not needed for since is always nonnegative..

8 Take a note: nth Root of nth-Powers
When a value has an exponent of n and you take the nth root you will get the value back again .. ... when a is positive (or zero): (i.e. for a ≥ 0) Example: ... or when the exponent is odd: (i.e. when n is odd Example: ... but when a is negative and the exponent is even you get this: ... so we have: (when n is even) (Note: |a| means the absolute value of a, in other words any negative becomes a positive Example:

9 Your turn Answers

10 Your turn again Find the two real solutions for each equation Answers:

11 Problem 4: Using a Radical Expression
The velocity of a falling object can be found using the formula where v is the velocity (in feet per second) and h is the distance the object has already fallen. What is the velocity of the object after 10 ft fall? How much does the velocity increase if the object falls 20 ft rather than 10 ft Answers: b) a)

12 Your turn: The speed s in meters per second of a car leaving a skid mark d meters long after the brakes are applied is given by the formula If you measure a skid mark and it is 180 m long, how fast was the car going before it braked? Answer: about 52 m/s

13 Classwork odd Homework even
TB pgs Exercises 10-48


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