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

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

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 ONE 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

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

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

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

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

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?

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

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

Inverse Variation You try! 1. Y varies inversely as X. When y = 25, x = 4.5. 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?

ALL DONE