Polynomials - Exponent Properties

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1 5.1 Polynomials - Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with expoenents can often e simplified using a few asic exponent properties. Exponents represent repeated multiplication. We will use this fact to discover the important properties. World View Note: The word exponent comes from the Latin expo meaning out of and ponere meaning place. While there is some deate, it seems that the Baylonians living in Iraq were the first to do work with exponents (dating ack to the 2rd century BC or earlier!) Example 1. a a 2 (aaa)(aa) a 5 Expand exponents to multiplication prolem Now we have 5a s eing multiplied together A quicker method to arrive at our answer would have een to just add the exponents: a a 2 = a +2 = a 5 This is known as the product rule of exponents Product Rule of Exponents: a m a n = a m+n The product rule of exponents can e used to simplify many prolems. We will add the exponent on like variales. This is shown in the following examples Example 2. Example. 2 6 Same ase, add the exponents x y 5 z 5xy 2 z 10x 4 y 7 z 4 Multiply 2 5, add exponents onx, y and z Rather than multiplying, we will now try to divide with exponents Example 4. a 5 a 2 aaaaa aa aaa a Expand exponents Divide out two of thea s Convert to exponents 1

2 A quicker method to arrive at the solution would have een to just sutract the exponents, a5 a 2 =a 5 2 =a. This is known as the quotient rule of exponents. Quotient Rule of Exponents: am a n = am n The quotient rule of exponents can similarly e used to simplify exponent prolems y sutracting exponents on like variales. This is shown in the following examples. Example Same ase, sutract the exponents 7 8 Example 6. 5a 5 c 2 2a c Sutract exponents on a, and c 5 2 a2 2 c A third property we will look at will have an exponent prolem raised to a second exponent. This is investigated in the following example. Example 7. 2 ) a 2 a 2 a 2 a 6 This means we have a 2 three times Add exponents Our solution A quicker method to arrive at the solution would have een to just multiply the exponents, (a 2 ) = a 2 = a 6. This is known as the power of a power rule of exponents. Power of a Power Rule of Exponents: (a m ) n = a mn This property is often comined with two other properties which we will investigate now. Example 8. (a) (a)(a)(a) a This means we have (a) three times Three a s and three s can e written with exponents 2

3 A quicker method to arrive at the solution would have een to take the exponent of three and put it on each factor in parenthesis, (a) = a. This is known as the power of a product rule or exponents. Power of a Product Rule of Exponents: (a) m = a m m It is important to e careful to only use the power of a product rule with multiplication inside parenthesis. This property does NOT work if there is addition or sutraction. Warning 9. (a +) ma m + m These are NOT equal, eware of this error! Another property that is very similar to the power of a product rule is considered next. Example 10. ) This means we have the fraction three timse ) ) ) Multiply fractions across the top and ottom, using exponents a A quicker method to arrive at the solution would have een to put the exponent on every factor in oth the numerator and denominator, ) a =. This is known as the power of a quotient rule of exponents. ) m= a Power of a Quotient Rule of Exponents: m m The power of a power, product and quotient rules are often used together to simplify expressions. This is shown in the following examples. Example 11. (x yz 2 ) 4 Put the exponent of4on each factor, multiplying powers x 12 y 4 z 8 Our solution

4 Example 12. ( ) a 2 Put the exponent of2on each factor, multiplying powers c 8 d 5 a 6 2 c 8 d 10 As we multiply exponents its important to rememer these properties apply to exponents, not ases. An expressions such as 5 does not mean we multipy 5 y, rather we multiply 5 three times, = 125. This is shown in the next example. Example 1. (4x 2 y 5 ) Put the exponent ofon each factor, multiplying powers 4 x 6 y 15 Evaluate4 64x 6 y 15 In the previous example we did not put the on the 4 and multipy to get 12, this would have een incorrect. Never multipy a ase y the exponent. These properties pertain to exponents only, not ases. In this lesson we have discussed 5 different exponent properties. These rules are summarized in the following tale. Rules of Exponents Product Rule of Exponents Quotient Rule of Exponents Power of a Power Rule of Exponents Power of a Product Rule of Exponents Power of a Quotient Rule of Exponents a m a n = a m+n a m a = n am n (a m ) n = a mn (a) m = a m m ) m= a m These five properties are often mixed up in the same prolem. Often there is a it of flexiility as to which property is used first. However, order of operations still applies to a prolem. For this reason it is the suggestion of the auther to simplify inside any parenthesis first, then simplify any exponents (using power rules), and finally simplify any multiplication or division (using product and quotient rules). This is illustrated in the next few examples. Example 14. (4x y 5x 4 y 2 ) In parenthesis simplify using product rule, adding exponents (20x 7 y ) With power rules, put three on each factor, multiplying exponents 20 x 21 y 9 Evaluate x 21 y 9 m 4

5 Example 15. 7a (2a 4 ) 7a (8a 12 ) 56a 15 Parenthesis are already simplified, next use power rules Using product rule, add exponents and multiply numers Example 16. a 10a 4 2a 4 2 0a 7 4 2a a 2 Simplify numerator with product rule, adding exponents Now use the quotient rule to sutract exponents Example 17. m 8 n 12 (m 2 n ) Use power rule in denominator m 8 n 12 m 6 n 9 m 2 n Use quotient rule Our solution Example 18. ( ) a 2 (2a 4 2 ) 2 6a 5 7 Simplify inside parenthesis first, using power rule in numerator ( ) a 2 (8a ) 6a 5 7 Simplify numerator using product rule ( ) 24a a 5 7 Simplify using the quotient rule ( 4a 8 ) 2 Now that the parenthesis are simplified, use the power rules 16a 16 2 Clearly these prolems can quickly ecome quite involved. Rememer to follow order of operations as a guide, simplify inside parenthesis first, then power rules, then product and quotient rules. Beginning and Intermediate Algera y Tyler Wallace is licensed under a Creative Commons Attriution.0 Unported License. ( 5

6 5.1 Practice - Exponent Properties Simplify. 1) ) ) m 4mn 7) 2m 4 n 2 4nm 2 9) ( ) 4 11) (4 4 ) 2 1) (2u v 2 ) 2 15) (2a 4 ) 4 17) ) 2 21) nm2 n 2) 4x y 4 xy 25) (x y 4 2x 2 y ) 2 27) 2x(x 4 y 4 ) 4 29) 1) ) 5) 2x 7 y 5 x y 4x 2 y ( (2x) x ) 2 ( 2y 17 (2x 2 y 4 ) 4 ) ( 2m n4 2m 4 n 4 mn 4 ) 7) 2xy5 2x 2 y 2xy 4 y 9) q r 2 (2p 2 q 2 r ) 2 2p ( ) zy z 41) x 4 y 4 4 x y z 4) 2x2 y 2 z 6 2zx 2 y 2 (x 2 z ) 2 2) ) 2 6) x 4x 2 8) x 2 y 4 xy 2 10) (4 ) 4 12) ( 2 ) 14) (xy) 16) (2xy) 4 18) 7 20) 4 22) x2 y 4 4xy 24) xy 4xy 26) (u 2 v 2 2u 4 ) 28) vu5 2v uv 2 2u v 0) 2a7 2 4 a 2 a 4 2) 2a2 2 a 7 (a 4 ) 2 4) yx2 (y 4 ) 2 2y 4 6) n (n 4 ) 2 2mn 8) (2y x 2 ) 2 2x 2 y 4 x 2 40) 2x4 y 5 2z 10 x 2 y 7 (xy 2 z 2 ) 4 ( ) 2q 42) p r 4 2p 4 (qrp ) 2 Beginning and Intermediate Algera y Tyler Wallace is licensed under a Creative Commons Attriution.0 Unported License. ( 6

7 5.1 1) 4 9 2) 4 7 ) 2 4 4) 6 5) 12m 2 n 6) 12x 7) 8m 6 n 8) x y 6 9) 12 10) ) ) 6 1) 4u 6 v 4 14) x y 15) 16a 16 16) 16x 4 y 4 Answers to Exponent Properties 17) ) 4 19) 20) 21) m 2 22) xy 4 2) 4x2 y 24) y2 4 25) 4x 10 y 14 26) 8u 18 v 6 27) 2x 17 y 16 28) uv 29) x2 y 6 0) 4a2 1) 64 2) 2a ) y 512x 24 4) y5 x 2 2 5) 64m 12 n 12 6) n10 2m 7) 2x 2 y 8) 2y 2 9) 2q 7 r 8 p 40) 4x 2 y 4 z 2 41) x 4 y 16 z 4 42) 256q 4 r 8 4) 4y 4 z Beginning and Intermediate Algera y Tyler Wallace is licensed under a Creative Commons Attriution.0 Unported License. ( 7