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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

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….

- 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 number line

Example number line

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

Example set of

Practice set of

- Ex. 3.2 (p. 211) #1-20
#1-2, 5-24

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.

Example set of

- 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.

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

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

- Ex. 4.3 (p. 217) #1-22

4. radicand as a product of 2 factors, one of which is the greatest perfect 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 and whole number exponents
- Example

- 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

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 .

- To give meaning to a power such as 8 and whole number exponents2/3, we extend the exponent law.
- m and n are rational numbers

- 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.

- Quick Tricks and whole number exponents

Example and whole number exponents

Example and whole number exponents

Example and whole number exponents

Practice and whole number exponents

- Ex. 4.4 (p. 227) #1-20
#1-2, 4-22

5. and whole number exponentsNegative Exponents

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

Example and whole number exponents

- The same rules apply for negative rational exponents. and whole number exponents

Example and whole number exponents

Example and whole number exponents

Practice and whole number exponents

- Ex. 4.5 (p. 233) #1-18
#3-21

6. and whole number exponentsOBEY the Laws of Exponents

Laws of Exponents and whole number exponents:

- Construct Understanding and whole number exponents 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

Examples contain rational bases

Examples contain rational bases

Examples contain rational bases

Examples contain rational bases

Practice contain rational bases

- Ex. 4.6 (p. 241) #1-19, 21
#1-2, 9-24

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