# B. Powers of Roots - PowerPoint PPT Presentation

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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

### 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

### 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….

• 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

### Example

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

### Practice

• 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.

### Example

• 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.

### Practice

• 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

### Example

• 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

### Practice

• 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

### Example

• The same rules apply for negative rational exponents.

### Practice

• Ex. 4.5 (p. 233) #1-18

#3-21

### 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

### Practice

• Ex. 4.6 (p. 241) #1-19, 21

#1-2, 9-24