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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

Index tells you what root to take

  • Radicand is what you are taking the root of
  • 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

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
  • Ex. 3.2 (p. 211) #1-20

#1-2, 5-24


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

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

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.
  • Ex. 4.4 (p. 227) #1-20

#1-2, 4-22

  • Ex. 4.5 (p. 233) #1-18



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