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B. Powers of Roots. Math 10: Foundations and Pre-Calculus. Key Terms:. Find the definition of each of the following terms : Irrational Number Real Number Entire Radical Mixed radical. 1. Remembering how to Estimate Roots. Index tells you what root to take

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B. Powers of Roots

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B powers of roots

B. Powers of Roots

Math 10: Foundations and Pre-Calculus


Key terms

Key Terms:

  • Find the definition of each of the following terms:

  • Irrational Number

  • Real Number

  • Entire Radical

  • Mixed radical


1 remembering how to estimate roots

1. Remembering how to Estimate Roots


B powers of roots

  • Index tells you what root to take

  • Radicand is what you are taking the root of


B powers of roots

  • Construct Understanding p. 205


Practice

Practice

  • Ex. 4.1 (p. 206) #1-6


2 what are irrational numbers

2. What are Irrational Numbers?

  • The formulas for the are of a circle and circumference of a circle involve π, which is not a rational number

  • It is not rational because it cannot be expressed as a quotient of integers


B powers of roots

  • π=3.14…..do you know what comes next?

  • π=3.14159265358979323846264383279….


B powers of roots

  • What are other non-rational numbers?


B powers of roots

  • Construct Understanding p. 207


B powers of roots

  • Radicals that are square roots of perfect squares, cube roots of perfect squares, and so on are rational numbers

  • Rational numbers have decimal representations that either terminate or repeat


B powers of roots

  • Irrational Numbers, cannot be written in the form m/n, where m and n are integers and n≠0

  • The decimal representation of an irrational numbers neither terminates nor repeats


B powers of roots

  • When an irrational number is written as a radical, the radical is the exact value of the irrational number

  • Examples


B powers of roots

  • We can use the square root and cube root buttons on our calculator to determine the approximate values of these irrational numbers

  • Examples


B powers of roots

  • We can approximate the location of an irrational number on a number line

  • If we don’t have a calculator, we use perfect powers to estimate the value


Example

Example


Example1

Example


B powers of roots

  • Together, the rational numbers and irrational numbers form a set of real numbers


Example2

Example


Practice1

Practice

  • Ex. 3.2 (p. 211) #1-20

    #1-2, 5-24


3 what is the difference between mixed and entire radicals

3. What is the Difference between Mixed and Entire Radicals


B powers of roots

  • We can use this property to simplify roots that are not perfect squares, cubes, etc, but have factors that are perfect squares, cubes, etc.


Example3

Example


B powers of roots

  • Some numbers, such as 200, have more than one perfect square factor

  • The factors of 200 are:

  • 1,2,3,4,5,8,19,20,25,40,50,100,200


B powers of roots

  • To write a radical of index n in simplest form, we write the radicand as a product of 2 factors, one of which is the greatest perfect nth power.


Example4

Example


Practice2

Practice

  • Ex. 4.3 (p. 217) #1-22


4 fractional exponents and radical relationships

4. Fractional Exponents and Radical Relationships

  • Construct Understanding p. 222


B powers of roots

  • In grade 9, you learned that for powers with variable bases and whole number exponents


B powers of roots

  • We can extend this law to powers with fractional exponents

  • Example


B powers of roots

  • So


B powers of roots

  • Raising a number to the exponent ½ is equivalent to taking the square root of the number

  • Raising to the exponent 1/3 is equivalent to taking the cube root


Example5

Example


B powers of roots

  • A fraction can be written as a terminating or repeating decimal, so we can interpret powers with decimal exponents.

  • For example, 0.2=1/5


B powers of roots

  • So 320.2 = 321/5

  • Prove it on you calculator .


B powers of roots

  • To give meaning to a power such as 82/3, we extend the exponent law.

  • m and n are rational numbers


B powers of roots

  • example


B powers of roots

  • These examples illustrate that numerator of a fractional exponent represents a power and the denominator represents a root.

  • The root and power can be evaluated in any order.


B powers of roots

  • Quick Tricks


Example6

Example


Example7

Example


Example8

Example


Practice3

Practice

  • Ex. 4.4 (p. 227) #1-20

    #1-2, 4-22


5 negative exponents

5. Negative Exponents


B powers of roots

  • Reciprocals are simply fractions that are the flip of each other


Example9

Example


B powers of roots

  • The same rules apply for negative rational exponents.


Example10

Example


Example11

Example


Practice4

Practice

  • Ex. 4.5 (p. 233) #1-18

    #3-21


6 obey the laws of exponents

6. OBEY the Laws of Exponents


Laws of exponents

Laws of Exponents:


B powers of roots

  • Construct Understanding p. 237


B powers of roots

  • We can use the exponent laws to simplify expressions that contain rational bases

  • It is conventional to write a simplified power with a positive exponent


Examples

Examples


Examples1

Examples


Examples2

Examples


Examples3

Examples


Practice5

Practice

  • Ex. 4.6 (p. 241) #1-19, 21

    #1-2, 9-24


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