1. Chapter 1B Powers
This section is not in the textbook and will mostly be a review from Math 10
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2. Review of Powers/Exponents
2) 83 = 8 x 8 x 8 = 512 3) 4) 5)
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3. Review of Roots (also called radicals)
Taking the root of a number is the inverse of an exponent. For example:
Since 32 = 9, 3 is the square root of 9 and thus
Since 33 = 27, is the cube root of 27 and thus
Since 34 = 81, is the cube root of 27 and thus
Types of Radicals
Laws of Radicals
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4. Converting Entire Radicals to Mixed Radicals
Example
Factors of 24 1 24 2 12 3 8 4 6
Step 1: determine all the factors of the radicand
Step 2: find which of the factors are perfect squares?
Step 3: Rewrite the radicand as a product of the factors which include the perfect square
Step 4: use the Root of a Product Law to separate into two different parts
Step 5: √4 = 2, Rewrite the equation
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5. Converting Entire Radicals to Mixed Radicals
Factors of 72 1 72 2 36 3 24 4 18 6 12 8 9
Example
When we have multiple perfect squares, choose the largest so 36
Example
Factors of 70 1 70 2 35 7 10
Since we have no perfect squares we cannot express as a mixed radical
Example
Since the index is 3 we need to find a factor with a perfect cube. 27 = 33
Factors of 54 1 54 2 27 3 18 6 7
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6. Converting Mixed Radicals to Entire Radicals
Example
Square and take the square root of the number outside the radical
Example
Take the cube and take the cube root of the number outside the radical
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7. Estimating Roots Estimate the value of
Write 2 consecutive perfect squares close to 17: = 16 & 52 = so the root will be between 4 and 5, but much closer to 4. Use a number line to evaluate further 42
≈4.1252
≈4.252
≈4.52 52 16
17.125
18.25
20.5 25
Divide the number line in half, quarters and then 8ths if necessary, until you have a value you are confident with
So we can see from the number line we created that ≈ So is close to a value of 4.1
Note “≈” means approximately. This method gives us a very rough estimate for the value of a root. As we look at cube roots, fourth roots ect. This method becomes even less accurate. When we get to even higher roots (think 15th root) this method completely falls apart. A better method will be taught in calculus
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8. Ordering a Set of Irrational Numbers from Least to Greatest
1) Rewrite the following list of numbers in order from least to greatest w/o using a calculator
Since every number in the list has an index of 2 or appears as a whole number we can square each term to better visual the order
If a > 0, then when a > b, a2 > b2 and if a < b then a2 < b2. So we can square each terms to determine the order of the terms.
= = = = = = 5
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9. Ordering a Set of Irrational Numbers from Least to Greatest
1) Without using a calculator determine which number is larger
Estimate both roots to determine which value is larger
Note that ≠ 17.5 and ≠ However the values are somewhat close and work as a very rough approximation
Start with
23 = 8 & = 27
Closer to 2 than 3 23
≈2.253
≈2.53 33 8
12.75
17.5 27
Since we know that 13 is slightly larger than we can estimate that will be somewhat close to 2.3
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10. Ordering a Set of Irrational Numbers from Least to Greatest
1) Without using a calculator determine which number is larger
24 = 16 & 34 = 81
Also closer to 2 than 3
Estimate the value of 24
≈2.254
≈2.54 34 16
≈27
32.25
48.5 81
Estimation: which is very close to the estimation of however,
Since 27 is closer to 2 on this number line than 13 was on the previous slide so we know that:
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11. Powers With Rational Exponents
A rational exponent refers to a fraction as a power. Rational exponents can be rewritten as a radical expression (√ )
Provided that m and n are integers with no common factors, and n must be greater than 1 (n > 1)
Why do you think these conditions exist?
Positive Exponent Examples 1) 2) 3)
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12. Powers With Negative Rational Exponents
Negative Exponent Examples 1)
Rationalize the denominator to simplify 2) 3)
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13. Exponent Laws
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14. Exponents Laws
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15. Order of Operations
Any expression in math which contains more than one operation, must follow the order of operations. Any operation that is the same …
Can be completed in the same step of the solution
Must be completed from left to right as we solve the problem
If the “order of operations” are not applied, or not applied correctly, the answer will usually be incorrect
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16. Order of Operations - Practice Problem
(7 - 4)2 + 4 • 21 ÷ 3 - 2
Brackets ( ) (7 - 4)2 + 4 • 21 ÷ 3 - 2
Also called parenthesis (3)2 + 4 • 21 ÷ 3 - 2
Exponents (3)2 + 4 • 21 ÷ 3 - 2
Also called powers • 21 ÷ 3 - 2
Multiple - Divide • 21 ÷ 3 - 2
Can happen in either order ÷ 3 - 2
Solve from left to right ÷ 3 - 2
Add - Subtract
can happen in either order = 35
Solve from left to right
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17. Assignment Suggested Questions
The content on this section is not found in the textbook. The practice problems will be found on the class handout. This content will be on the chapter 1 exam
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