LAWS OF EXPONENTS

22 = 2 X 2 X 2 X 2 Basic Terminology Exponent – In exponential notation, the number of times the base is used as a factor. Base – In exponential notation, the number or variable that undergoes repeated MULTIPLICATION. EXPONENT 22 = 2 X 2 X 2 X 2 BASE Review the vocabulary. The base and the exponent together is called the power.

IMPORTANT EXAMPLES -34 means –(3 x 3 x 3 x 3) = -81 Very important to stress that the negative sign only goes with the number if it is included in the parentheses. (-3)3 means (–3) x (–3) x (–3) = -27

Try these: – 24 – 52 (– 5)2

x5 = (x)(x)(x)(x)(x) y3 = (y)(y)(y) Variable Expressions Explain that using parentheses is the same as multiplication.

Substitution and Evaluating STEPS Write out the original problem. Show the substitution with parentheses. Work out the problem. Solve x3 if x = 4 x3 43 = 64

Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3

Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3

Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Walk the partiicpants through each step. Step 3

MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base. a3 x a4 = (a x a x a) x (a x a x a x a) = a7 d3d6 = (d x d x d) x (d x d x d x d x d x d) = a3+6 = a9 23 x 25 = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 2) = 28 = 256 Ask the participants what they notice about the exponents in the problems and the exponent of the answer. Let them discover that when you multiply powers with the same base you will add the exponents.

MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base. (52)3 = (52)(52)(52) = 52+2+2 = 56 =b8 (b2)4 = (b2)(b2)(b2)(b2) = b2+2+2+2 = b(2x4) Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times. Do several more examples ((34)5 = 320, (y2)7 = y14 Help the participants discover that when you raise a power to a power you multiply the exponents. (43)5 = 43x5 = 415

MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (-6 x 5)2 = (-6)2 x (5)2= Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times. Do several more examples ((34)5 = 320, (y2)7 = y14 Help the participants discover that when you raise a power to a power you multiply the exponents. 36 x 25 = 900 900

MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (5xy)3 = Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times. Do several more examples ((34)5 = 320, (y2)7 = y14 Help the participants discover that when you raise a power to a power you multiply the exponents. (53)(x3)(y3)= 125x3y3

MULTIPLICATION PROPERTIES POWER OF PRODUCT (4a2)3 x a5= (43)(a2)3 x a5= (64)(a2x3) x a5= Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times. Do several more examples ((34)5 = 320, (y2)7 = y14 Help the participants discover that when you raise a power to a power you multiply the exponents. (64)(a6) x a5= (64)(a6+5) 64a11

1. (2xy)3 x2 2. (2x3y)4 x4y5 3. (3x2y3)2 (4x2)2 4. (2x)2 (3x+y)2 Simplify each expression. 1. (2xy)3 x2 2. (2x3y)4 x4y5 3. (3x2y3)2 (4x2)2 4. (2x)2 (3x+y)2

MULTIPLICATION PROPERTIES PRODUCT OF POWERS add the exponents SUMMARY ya x yb = ya+b PRODUCT OF POWERS add the exponents POWER TO A POWER multiply the exponents (ma)b = ma•b Review POWER OF PRODUCT distribute the exponents (cd)a = cada

ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 24 = 2 x 2 x 2 x 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 =2 2 ÷ 2 = 1 Dividing by the base number 23 = 2 x 2 x 2 = 8 22 = 2 x 2 = 4 Have the participants do this themselves with other base numbers. Then ask them what would happen if the kept dividing the answers by the base number. Example of this on the next slide. 21 = 2 20 = 1

ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 =2 2 ÷ 2 = 1 Dividing by the base number 33 = 27 32 = 9 31 = 3 30 = 1 3-1= Have the participants try this with another problem. 54 3-2 = 3-3 =

ZERO AND NEGATIVE EXPONENTS Explain in this example that the -2 is the exponent for the x but in the next example the -2 exponent applies to the 2 and the x because of the parenthesis.

ZERO AND NEGATIVE EXPONENTS Explain in the previous example that the -2 is the exponent for the x but in this example the -2 exponent applies to the 2 and the x because of the parenthesis.

Simplify each expression.

DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base. Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base. Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base. Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

DIVISION PROPERTIES QUOTIENT OF POWERS The exponent 4 is distributed to both the numerator and denominator.

ZERO, NEGATIVE, AND DIVISION PROPERTIES REVIEW ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power Quotient of powers Negative Exponents Power of a quotient Review the properties