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B. Powers of Roots

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B. Powers of Roots

Math 10: Foundations and Pre-Calculus

- 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

- Construct Understanding p. 205

- Ex. 4.1 (p. 206) #1-6

- 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

- 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

- 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

- 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
#3-21

- 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