Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Exponential Laws and Equations The Introduction to Exponents! A long time ago XXXX was written X X X X . This became so long and complicated that X X X X was turned into the notation X4, which means X multiplied by itself four times. X4 is called an exponential notation. Exponents are represented in this form. (n) X Exponent Base Exponent: The number of times the base is multiplied by its self. Base: The number that is being multiplied. X0 = 1 X1 = X X2 = X X X3 = XXX And so on……. To change a number into a exponent: l log number log new base = New Exponent Example: Log27 = 3 Log3 27 can be represented as 33 33 = (3)(3)(3) = (9)(3) = 27 ** Sometimes the operator ^ is used in calculations to represent an exponent Example: Xn = X^n When using large numbers such as 1000000, they can get large and messy. We can simplify them by using exponents. Example: 1 000 000 = 101010101010 = 106 Negative Bases: Negatives being applied to the bases must be watched carefully. If the negative is not in brackets with the base the negative will also be applied to the answer. Example: -25 = -(22222) = -32 If the negative being applied to the base is in brackets and the exponent is even than the negative will not apply to the answer. Example: (-2)4= (-2)(-2)(-2)(-2) = 16 If the negative being applied to the base is in brackets and the exponent is odd than the negative will apply to the answer. Example: (-2)5 = (-2)(-2)(-2)(-2)(-2) = -32 Exponent Laws! Rule 1: If the bases are the same and multiplied, you can add their exponents Rule 2: If the bases are the same and divided, you can subtract their exponents Rule 3: If a power is raised to a further power, you must multiply the exponents If you have different bases, multiplied with each other, raised to a power, you must apply rule 3 for each of the bases! Remove the ( )'s! Rule 4: Rule 5: Same goes for division! The exponent goes for the top and the bottom (the numerator and the denominator) Rule 6: or Rules and Laws Explained: Negative exponents: (if you are negative on top rather go underground) or (if you buries underground get on top and be positive P Rule 1: Negative Exponents A number in exponential notation can be expressed as a reciprocal. A reciprocal is 1 over the base to the opposite sign on the exponent. Example: Xn = 1/x-n and X^-n = 1/x^n 3-3 = 1/33 = 1/(3)(3)(3) = 27 33 = (3)(3)(3) = 27 Rule 2: Multiplication If the bases are the same and you are multiplying then you can add the exponents. Example: X2 + X2 = X2+2 = X4 26 + 22 = 26+2 = 28 = (2)(2)(2)(2)(2)(2)(2)(2) = 256 Rule 3: Division If the bases are the same and you are dividing the you can subtract the exponents. Example: X / X2 = X3-2 = X1 =X 26 / 24 = 26-4 = 22 =(2)(2) =4 Rule 4: Raising a power to an exponent When you are raising a power to an exponent keep the base and multiply the exponents. Example: (r4)5 = r4*5 = r20 (52)3 = 52*3 = 56 = 15625 Rule 5: Different Bases If you have two bases that are being multiplied or divided and raised to a power you have to use rule 4 for both of the bases. Then you remove the brackets. Example: (Multiplying) (XY)3 = X3 Y3 (3*4)2 = 32 42 = (3)(3) (4)(4) = (9) (16) = 144 (Dividing) (X/Y)2 = X2 / Y2 (4/2)2 = 42/ 22 = (4)(4) / (2)(2) = 16 / 4 =4 Questions To Try: 1. 2. 3. 64x6y9 25x6y12 416x4y12 3 –5a-3 / a2 5. 6x12y3z2 / -6x6z2 6. (-2a3b2)2 / 4a3b2 7. –(-xy)4 8. (17x5y9z8)2 9. (xa)a+5 10. 23x2y3(22x4y3)(2xy2z) 11. 32(x4)(y3)(x8)(y2)(22) 12. x18y9z10 – x6z5 13. 2xy3(4x2y) / (2xy2)3 14. 4a3b4c6 / 12a2b4c0 15. (-5x5y9)2(-5x8y2)3 4. Answers: 1. 3 64x6y9 (43x6y9)1/3 4x2y9 The root becomes the denominator. To take the cubed root of am exponent, divide by 3. 2. 25x6y12 (52x6y12)1/2 5x2y6 The root becomes the denominator. To take the square root of an exponent, divide by 2. 3. 464x4y12 (43x4y12)1/4 43/4xy3 The root becomes the denominator. To take the fourth root of an exponent divide by 2. 4. –5a-3 / a2 -5a-3-2 -5a-5 -5 / a5 The base is the same so subtract the exponents. The negative exponent becomes positive by flipping the base. 5. 6x12y3z2 / -6x6z2 -6x12-6y3z2-2 -6x6y3 A (+) divided by a (–) equals a (-). The exponents on the like bases are subtracted. 6. (-2a3b2)2 / 4a3b2 4a6b4 / 4a3b2 4a6-3 b4-2 4a3b2 The terms in the brackets are squared. The exponents on the bases that are the same are subtracted. 7. –(-xy)4 -(-x4y4) x4y4 The brackets are removed by multiplying the exponent on the terms by the exponent on the bracket. A (-) times a (-) is a (+) 8. (17x5y9z8)2 289x10y18z16 The brackets are removed by multiplying the exponents on the terms by the one on the bracket. 9. (xa)a+5 xa2+5a A*A=A2 5*A = 5A 10. 23x2y3(22x4y3)(2xy2z) 8(4)(2)x2+4+1y3+8+2z 64x7y8z The numbers are multiplied by themselves. The terms are being multiplied so exponents of the same base are added. Test Questions: –(-x2y3)6 (xaya) a+1 16a4b6c8 x7y3z2 x3y3z2 –6a -4 a2 6. 23x2y2 ( 22x4y3)( 2xy2z) 7. 2xy3(4x2y) (2xy2)3 8. 4a3b4c6 12a2b4c0 9. 3 27x6y12 10. (3x6y9)2 – (6x6y9)2 1. 2. 3. 4. 5. Test Answers: 1. –(-x2y3)6 = -(-x12y18) = x12y18 2. (xaya)a+6 = x a2+6a y a2+6a 3. 16a4b6c8 = (24a4b6c8) ½ = 4a2b3c4 4. x7y3z2 x3y3z2 = x 7-3 y 3-2 z 2-2 = x4 y 5. –6a-4 a2 = -6a-4-2 =-6a-6 = -6 a6 6. 23x2y3 (22x4y3)( 2xy2z) = 8(4)(2) x 2+4+1 y 2+3+3 z = 64 x7y8z 7. 2xy2(4x2y) (2xy2)3 = 8x3y4 23x3y6 =8-8 x 3-3 y 4-6 = y –2 =1 y2 8. 4a3b4c6 12a2b4c0 = 1/3 a 3-2 b 4-4 c 6-0 = 1/3 ac6 = ac6 3 9. 3 27x6y12 = (33 x6y12) 1/3 = 3x2y6 10. (3x6y9)2 – (6x6y9)2 = 32x12y18 – 62x12y18 =9-36 x 12-12 y 18-18 = -27 = 1 27 Links To Exponent WebPages: 1) http://www.sosmath.com/algebra/logs/log3/log3.html 2) http://www.math.com/school/subject2/lessons/S2U2L2GL.html 3) http://www.edhelper.com/exponents.htm Bibliography: Algebra Basics, “Exponents”, 2000, http://www.math.com/school/subject2/lessons/S2U2L2GL.html Ed Helper.com, “Exponent Worksheets” http://www.edhelper.com/exponents.htm “Exponential and Surd Worksheet Summary” http://www.easymaths.org/Gr%2011/Algebra/exsitemap.htm Henderson, Douglas, Starting Points In Math 10, 1982, Canada: Ginn and Company Educational Publishers Marcus, Nancy, “Exponential Rules, 2002, USA: Math Medics http://www.sosmath.com/algebra/logs/log3/log3.html Math League Multimedia, “Decimals, Whole Numbers, and Exponents”, 2002, http://www.mathleague.com/help/decwholeexp/decwholeexp.htm