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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

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

  • Radicand is what you are taking the root of

  • Construct Understanding p. 205


  • Ex. 4.1 (p. 206) #1-6

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….

  • What are other non-rational numbers?

  • Construct Understanding p. 207

  • 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

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

  • Examples

  • 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



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



  • Ex. 3.2 (p. 211) #1-20

    #1-2, 5-24

3. What is the Difference between Mixed and Entire Radicals

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


  • 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

  • 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.



  • Ex. 4.3 (p. 217) #1-22

4. Fractional Exponents and Radical Relationships

  • Construct Understanding p. 222

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

  • We can extend this law to powers with fractional exponents

  • Example

  • So

  • 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


  • 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

  • So 320.2 = 321/5

  • Prove it on you calculator .

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

  • m and n are rational numbers

  • example

  • 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.

  • Quick Tricks





  • Ex. 4.4 (p. 227) #1-20

    #1-2, 4-22

5. Negative Exponents

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


  • The same rules apply for negative rational exponents.




  • Ex. 4.5 (p. 233) #1-18


6. OBEY the Laws of Exponents

Laws of Exponents:

  • Construct Understanding p. 237

  • 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






  • Ex. 4.6 (p. 241) #1-19, 21

    #1-2, 9-24

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