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00 that week Percentages (including percent increase and decrease) To find the amount of sales tax on an item, change the percent of sales tax into an equivalent decimal number. Then multiply the decimal number times the price of the object to find the sales tax. The total cost of an item will be the price of the item plus the sales tax. Example: A guitar costs $120 plus 7% sales tax. How much are the sales tax and the total bill? 40 MIDDLE SCHOOL MATHEMATICS 26 TEACHER CERTIFICATION STUDY GUIDE Example: An item that is on sale costs $18.

The Laws of Exponents The exponent form is a method to write repeated multiplication. Basic form: b n , where b is called the base and n is the exponent. b and n are both real numbers. b n implies that the base b is multiplied by itself n times. Examples: 1) 34 = 3 × 3 × 3 × 3 = 81 where 3 is the base and 4 is the exponent. 3 2) 2 = 2 × 2 × 2 = 8 − 4 − − − − 3) ( 2) = ( 2) × ( 2) × ( 2) × ( 2) = 16 − 4 − − 4) 2 = (2 × 2 × 2 × 2) = 16 MIDDLE SCHOOL MATHEMATICS 41 TEACHER CERTIFICATION STUDY GUIDE The following are basic rules for exponents: = a1 a= for all values of a; thus 171 17 = b0 1= for all values of b; thus 240 1 = 10n 1= with n zeros; thus 106 1,000,000 Example: 107 = 10,000,000 x2 is read, “x squared” y3 is read, “y cubed” For real numbers m and n, and for nonzero ‘a’, Product Rule 1) a m ⋅ a n = a( m + n ) Example: (34)(35) = 39 2) am ⋅ am = (ab )m Example: (42)(52) = 202 3) (am )n = amn Example: (23)2 = 26 Quotient Rule 4) am = a( m − n ) n a Example: 25 ÷ 23 = 22 5) a − m = 1 am Example: 2–2 = 1 22 MIDDLE SCHOOL MATHEMATICS 42 TEACHER CERTIFICATION STUDY GUIDE Samples of simplifying expressions with exponents: (−2)3 = −8 -23 = −1(23 ) = −8 (−2)4 = 16 -24 = −1(24 ) = −16 Note change of sign.

We say that ‘6 squared is 36’, or ‘6 to the power 2 is 36’. 25 = 2× 2 × 2 × 2 × 2 = 32. We say that ‘2 to the power 5 is 32’. Square roots A square root is a number that gives the original number when squared. ” For example, 9 = 3 since ( 3 ) = 9 . 2 When 5 is squared, we get 25: 52 = 25. The reverse of this process is called finding a square root. The square root of 25 is 5. This is written as 2 25 = 5 or 25 = 5 . e. (−5)2 = 25. This means that the number 25 has another square root, −5. Therefore, the two square roots of 25 are +5 and –5.