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MPM2D WORKING WITH EXPONENTS Exponents are a simpler way of showing a repeated multiplication. Powers are products of equal factors. They can be written in expanded form or exponential form. 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 can also be written as 38 the entire expression is called a POWER 38 the number 8 is the EXPONENT, it tells us the number of times the base has been repeated the number 3 is the BASE, it tells us the number that is being multiplied by itself THE EXPONENT LAWS Example 1: Multiply (4)3 (4)2 if we expand (4)3(4)2 Notice - the bases are the same it means (4 x 4 x 4) (4 x 4) = 45 Example 2: Multiply (x)4 (x)2 if we expand = (x)(x)(x)(x) (x)(x) Example 3: Multiply (x2y3)(x3y4) if we expand = (x)(x)(y)(y)(y) = (x)6 (x)(x)(x)(y)(y)(y)(y) = (x)5(y)7 LAW #1: MULTIPLICATION OF POWERS To multiply powers with the same base, add the exponents and keep the base the same (x)m (x)n = x m+n Example 4: Divide (4)5 ÷ (4)2 Example 5: Divide (x4) ÷ (x)2 (4)( 4)( 4)( 4)( 4) (4)( 4) if we expand = reduce = (4)3 = ( x)( x)( x)( x) ( x)( x) = (x)2 Example 6: Divide (x3y4) ÷ (x2y2) if we expand = reduce = xy2 ( x)( x)( x)( y )( y )( y )( y ) ( x)( x)( y )( y ) LAW #2: DIVISION OF POWERS To divide powers, subtract the exponents and keep the base (x)m ÷ (x)n = x m-n Example 7: Expand (53)2 If we expand = (53)(53) = (x2y3)(x2y3)(x2y3) simplify = 56 = x6y9 Example 8: Expand (x2y3)3 3 Example 9: Expand If we expand simplify x2 2 y x 2 x 2 = 2 2 y y x6 = 6 y x 2 2 y POWER OF A QUOTIENT LAW POWER OF A POWER LAW To apply a power to a power, multiply the exponents and keep the base the same (xm)n = xmn To apply a power to a quotient, multiply the exponents in both the numerator and the denominator, keep the base the same. xm m y n x mn mn y ZERO AND NEGATIVE EXPONENTS We can use the patterns and the basic laws of exponents to understand the meaning of zero and negative exponents. 43 ÷ 4 3 Example 1: Simplify 43 ÷ 43 4 4 4 = 4 4 4 =1 43 ÷ 43 but also = 4 3-3 = 40 therefore, 40 = 1 Example 2: Simplify 42 ÷ 45 42 ÷ 45 42 ÷ 45 but also 4 4 4 4 4 4 4 1 = 4 4 4 1 = 3 4 = 4 2-5 = therefore, 4-3 = = 4 –3 1 43 NEGATIVE EXPONENTS Any power with a negative exponent can be expressed in an equivalent fraction form with a positive exponent 1 n x5-B-2 n x **Exponents change their sign if they move from numerator to denominator OR from denominator to numerator! Always try to express answers with positive exponents only!** Eg. 1 = 43 4 3 and x 2 y 3 5 6 = x y EXPONENT WORKSHEET Use the exponent laws we’ve learned to find the following: 1) State the following numbers as powers of 2 a) 16 _______ b) 2 d) (28)(27) _________ _______ c) 32 e) 2 x 29 ________ f) _________ 28 _________ 24 g) 4 x 32 _________ (hint: change to same base) h) 4 x 8 _________ 2) State the following numbers as powers of 3 a) 27 d) 34 _______ 35 x _________ b) 243 _________ c) 38 x 3 __________ 35 e) 2 3 32 f) 5 3 ________ __________ g) 27 x 81 _________ 3) Simplify the following expressions a) (3x ) 3 3 d) f) _________ 710 712 7 6 7 3 9 2 n 2 4 n1 8n b) 3x 6 4x ____________ 3 x2 c) 4 y __________ ( x m n )( x 2 m ) e) xn 3 ________ _____________ ____________ (hint: change to the same base) 4) Simplify the following expressions 2 x y 8 x y x y 2 a) 4 3 2 5 2 3 __________ b) 2 xy 4 15 x 2 y 3 3x 2 y 2 12 x 4 y 2 _____________ c) 2 n 4 n 1 8 3n 2 16 2 n 1 (HINT: express each with a base of 2) 5) State the following using only positive exponents a) x 8 ________ b) x 2 y 2 1 2 1 ________ e) g) 3 2 ________ h) (-3 x 2 y 2 )( 4 x 2 y 2 ) _________ d) 1 3 1 _________ _________ c) a 3 b 4 ________ 1 2 3 ________ f) 6) Evaluate the following a) 50 _______ 1 d) 2 3 4 f) 0 3 8 (express with same base) h) 5-1 + 5-2 (careful!) c) (3-2)-2 ________ 2 3 2 2 27 2 g) 2-1 + 3-1 b) 4-2 _________ _________ e) 2 5 __________