1. Unit 4: Exponents, radicals, and variation
Final Exam Review

2. Topics to cover Exponent Rules
Converting Radicals to Fractional Exponents
Converting Fractional Exponents to Radicals
Solving Rational Equations
Direct Variation
Inverse Variation

3. Exponent rules
Mathematical Expressions can be simplified used exponent rules
Here are all of the rules:
ADDING AND SUBTRACTING EXPRESSIONS
MULTIPLYING EXPRESSIONS
RAISING A POWER TO A POWER
DIVIDING EXPRESSIONS
NEGATIVE EXPONENTS
ZERO EXPONENTS

4. ADDING AND SUBTRACTING EXPRESSIONS
When you are adding and subtracting exponents, you must:
COMBINE LIKE TERMS only!
Make sure to DISTRIBUTE the NEGATIVE when subtracting
Example:
(4x2 + 9x – 6) + (7x2 – 2x – 1) = x2 + 7x – 7
(3x2 + 5x – 8) – (5x2 – 4x + 6) = x2 + 9x – 14

5. Multiplying expressions
When you are multiplying expressions
MULTIPLY the whole numbers
ADD the exponents
Example:
(4x3)(2x2) = x5
(-4x5)(3x2) = x7

6. Raising a power to a power
When you are raising a power to a power:
RAISE the whole numbers to the power
MULTIPLY the exponents
Example:
(5x2) = x8
(-3x6) = x18

7. Dividing expressions When you are dividing expressions:
DIVIDE the whole numbers
SUBTRACT the exponents
Example:
6𝑥 4 4𝑥 = 𝟑𝒙 𝟐 𝟐
8𝑥 3 𝑦 𝑥 8 𝑦 = 𝟒𝒚 𝟒 𝟓 𝒙 𝟓

8. Negative exponents When you have a negative exponent
MOVE the negative exponent “TO THE OTHER BUNK”, meaning, move it to the other side of the FRACTION
When you move it, change the exponent to a POSITIVE because now it’s “HAPPY”
Example:
𝑥 −5 𝑦 − = 𝒚 𝟐 𝒙 𝟓
𝑥 2 𝑦 −7 𝑥 −5 𝑦 = 𝒙 𝟐 𝒙 𝟓 𝒚 𝟒 𝒚 𝟕 = 𝒙 𝟕 𝒚 𝟏𝟏

9. Zero exponents When you have a zero exponent The answer is always ONE
Example:
(5x4y2) =
4𝑥 3 𝑦 2 6𝑥 −4 𝑦− =

10. Practice all exponent rules
(5x2 – 5x + 2) + (6x2 + 2x – 10)
(3x2 + 6x – 4) – (6x2 – 2x + 9)
(6x4)(5x2)
(4x2)3
𝟖𝐱 𝟓 𝐲 𝟐 𝟏𝟐𝐱 𝟑 𝐲 𝟗
𝟔𝐱 𝟓 𝐲 −𝟓 𝟖𝐱 −𝟐 𝐲 𝟑
(3x2y)0

11. Converting a radical into a fractional exponent
Parts of a radical
When converting a radical to a fractional exponent:
The power inside the radical becomes the NUMERATOR
The number in the INDEX becomes the DENOMINATOR
Example: 𝑥 3 = 𝑥 3 5

12. Converting a radical into a fractional exponent
Now try these:
7 𝑥 6
𝑥 3
6 (2𝑥) 11
3 2𝑥 4

13. Converting a fractional exponent into a radical
When converting a fractional exponent into a radical:
The numerator becomes the power INSIDE the radical
The denominator becomes the number in the INDEX
Example: 𝑥 = 𝑥 2

14. Converting a fractional exponent into a radical
Now try these:
𝑥 4 5
3𝑥 1 2
(6𝑥) 6 11
𝑥 5 4

15. Solving rational equations
When solving rational equations with 2 terms, you must CROSS MULTIPLY
Example: 2𝑥+1 20 = 𝑥 5
5(2x+1) = 20x
10x + 5 = 20x
5 = 10x
x = 1 2

16. Solving rational equations
You try:
Example: 𝟒𝒙−𝟐 𝟐 = 𝟒𝒙+𝟕 𝟓

17. Direct Variation
Direct Variation is a relationship between 2 variables when 1 variable INCREASES the other variable also INCREASES.
Equation: y = kx
K = CONSTANT

18. Direct Variation Example:
Y varies directly as X. When y = 27, x = 6. What does y equal when x = 10?
Y = kx  27 = k(6)  k = 4.5
Y = 4.5(10)  y = 45

19. Direct Variation You try!
1. Y varies directly as X. When y = 150, x = 35. What does y equal when x = 99?
2. The amount of money that student government makes selling homecoming t-shirts varies directly as the number of students who buy them. If SGA makes $350 when 25 people buy a shirt, how much money will they make if 55 people buy a shirt?

20. inverse Variation
Inverse Variation is a relationship between 2 variables when 1 variable INCREASES the other variable DECREASES.
Equation: 𝐲= 𝒌 𝒙
K = CONSTANT

21. Inverse Variation Example:
Y varies inversely as X. When y = 18, x = What does y equal when x = 12?
𝐲= 𝒌 𝒙  18 = 𝒌 𝟐  k = 36
𝐲= 𝟑𝟔 𝟏𝟐  y = 3

22. Inverse Variation You try!
1. Y varies inversely as X. When y = 25, x = What does x equal when y = 10?
2. The average speed that you drive varies inversely as the time it takes to travel. It takes Sally 230 minutes to drive to Disney World when she averages 60mph. How long will it take her friend Abby to travel the same distance to Disney if she averages 53mph?

23. ALL DONE