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

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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

Index tells you what root to take

• Radicand is what you are taking the root of
Practice
• 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….

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

• 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.

Practice
• 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 .
• 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.
Practice
• Ex. 4.4 (p. 227) #1-20

#1-2, 4-22

Practice
• Ex. 4.5 (p. 233) #1-18

#3-21

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
Practice
• Ex. 4.6 (p. 241) #1-19, 21

#1-2, 9-24