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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

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


  • Index tells you what root to take

  • Radicand is what you are taking the root of



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


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

  • π=3.14159265358979323846264383279….




  • 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


  • 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


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

  • Examples




Example
Example number line


Example1
Example number line



Example2
Example set of


Practice1
Practice set of

  • Ex. 3.2 (p. 211) #1-20

    #1-2, 5-24


3 what is the difference between mixed and entire radicals
3. set of What is the Difference between Mixed and Entire Radicals


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


Example3
Example set of




Example4
Example radicand as a product of 2 factors, one of which is the greatest perfect


Practice2
Practice radicand as a product of 2 factors, one of which is the greatest perfect

  • Ex. 4.3 (p. 217) #1-22


4 fractional exponents and radical relationships
4. radicand as a product of 2 factors, one of which is the greatest perfect Fractional Exponents and Radical Relationships

  • Construct Understanding p. 222




  • So and whole number exponents


  • Raising a number to the and whole number exponentsexponent 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 and whole number exponents


  • A fraction can be written as a and whole number exponentsterminating or repeating decimal, so we can interpret powers with decimal exponents.

  • For example, 0.2=1/5


  • So 32 and whole number exponents0.2 = 321/5

  • Prove it on you calculator .



  • example and whole number exponents


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

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



Example6
Example and whole number exponents


Example7
Example and whole number exponents


Example8
Example and whole number exponents


Practice3
Practice and whole number exponents

  • Ex. 4.4 (p. 227) #1-20

    #1-2, 4-22


5 negative exponents
5. and whole number exponentsNegative Exponents


  • Reciprocals and whole number exponents are simply fractions that are the flip of each other


Example9
Example and whole number exponents



Example10
Example and whole number exponents


Example11
Example and whole number exponents


Practice4
Practice and whole number exponents

  • Ex. 4.5 (p. 233) #1-18

    #3-21


6 obey the laws of exponents
6. and whole number exponentsOBEY the Laws of Exponents


Laws of exponents
Laws of Exponents and whole number exponents:




Examples
Examples contain rational bases


Examples1
Examples contain rational bases


Examples2
Examples contain rational bases


Examples3
Examples contain rational bases


Practice5
Practice contain rational bases

  • Ex. 4.6 (p. 241) #1-19, 21

    #1-2, 9-24


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