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Section 3.4 Division and Exponents 1 Division of Whole Numbers For any whole numbers r and s, with s ≠ 0, the quotient of r divided by s, written r ÷ s, is the whole number k, if it exists, such that r = s × k. The dividend is r, the divisor is s, and the quotient is k. 2 Show that 24 ÷ 6 = 4 using both of the following concepts of division. Measurement (Subtractive) Concept of Division How many groups of 6 are in 24? Sharing (Partitive) Concept of Division If we have 24 we want to divide into 6 groups, how many are in each group? 3 What division fact is illustrated? 0 3 6 9 12 4 How could you show 29 ÷ 7 using base 10 pieces • Subtractive: • Partitive: 5 Division Theorem For any whole numbers a and b with divisor b ≠ 0, there are whole numbers q (quotient) and r (remainder) such that a = bq + r and 0 ≤ r < b. Ex: 19 ÷ 5 (19 = a, and 5 = b); we can write: 19 = (5)(quotient) + remainder Or: 19 = (5)(3) + 4 6 Write the division as multiplication. 1) 102 ÷ 17 = 6 2) 546 ÷ 26 = 21 Write the multiplication as division. 1) 14 x 31 = 434 2) 45 x 7 = 315 7 Method of Equal Quotients The quotient of two numbers remains the same when both numbers are divided by the same number. Use the method of equal quotients to replace the divisor and the dividend with smaller numbers. Show the new quotient that replaces the original quotient. Repeat this process, if necessary, until you can mentally calculate the exact quotient. 1) 20 ÷ 10 2) 486 ÷ 18 8 Use compatible numbers to mentally estimate the quotient. 1) 92 ÷ 14 2) 489 ÷ 47 9 Laws of Exponents For any number a and all whole numbers m and n, except for the case where the base and exponents are both zero, n m n+ m n m n−m a ×a = a a ÷a = a for a ≠ 0 10 Compute the products and quotients. Leave in exponential form. 10 13 6 9 1) 6 × 6 2) 7 × 7 8 3) 5 ÷ 5 42 3 4) 17 ÷ 17 20 11 Order of Operations Please Excuse My Dear Aunt Sally PEMDAS How does this help us remember the order of operations? Evaluate the following expressions. 1) 2 x 8 – 3 x 5 2) (10 + 6) ÷(32 – (4 – 3)) 3) 32 ÷ 42 + (5 + 2)2 - 8 12