Unit 3: Exponents, radicals, and exponential equations Final Exam Review

Topics to cover Exponent Rules Converting Radicals to Fractional Exponents Converting Fractional Exponents to Radicals Exponential Growth and Decay Word Problems

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

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) = 11x2 + 7x – 7 (3x2 + 5x – 8) – (5x2 – 4x + 6) = -2x2 + 9x – 14

Multiplying expressions When you are multiplying expressions MULTIPLY the whole numbers ADD the exponents Example: (4x3)(2x2) = 8x5 (-4x5)(3x2) = -12x7

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)4 = 625x8 (-3x6)3 = -27x18

Dividing expressions When you are dividing expressions: DIVIDE the whole numbers SUBTRACT the exponents Example: 6𝑥 4 4𝑥 2 = 𝟑𝒙 𝟐 𝟐 8𝑥 3 𝑦 6 10𝑥 8 𝑦 2 = 𝟒𝒚 𝟒 𝟓 𝒙 𝟓

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 −2 = 𝒚 𝟐 𝒙 𝟓 𝑥 2 𝑦 −7 𝑥 −5 𝑦 4 = 𝒙 𝟐 𝒙 𝟓 𝒚 𝟒 𝒚 𝟕 = 𝒙 𝟕 𝒚 𝟏𝟏

Zero exponents When you have a zero exponent The answer is always ZERO Example: (5x4y2)0 = 1 4𝑥 3 𝑦 2 6𝑥 −4 𝑦− 2 0 = 1

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

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: 5 𝑥 3 = 𝑥 3 5

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

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 2 5 = 5 𝑥 2

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

Exponential growth and decay Exponential Functions can either represent GROWTH or DECAY Every function follows this formula: y = a bx a is the INITIAL value b is the GROWTH or DECAY rate If the problem is growth, use (1 + rate) for b If the problem is decay, use (1 – rate) for b x is the TIME

Exponential growth and decay Example Write the equation for this situation: The amount of movies made in 2015 was 1,255. The number is expected to increase by 2.1% every year. Answer: y = 1255(1 + 0.021)x

Exponential growth and decay Now try these Write an equation for these situations: The population of an ant colony with 5,056 members increases by 5.6% every year. The number of people who live in North Dakota (who currently has 739,482 people) decreases every year by 1.3%.

Word problems There are many real life situations that use exponential growth and decay. You can use these equations in order to predict outcomes in the future. In order to do this, use your calculator to put in the equation and use the table to find values.

Word problems Try this one: The model y = 604000(1 + 0.045)x represent the population of Washington DC after 1990. 1. Find the initial population 2. Is this a growth or decay problem? 3. Predict the population in 1995. 4. In what year will the population reach 1,000,000?

ALL DONE