Exponential Rules and How to Use Them Together

Transcription

1 Epoetial Rules ad How to Use Them Together Welcome back! Before we look at problems that ivolve the use of all of our epoetial rules, let s review the epoetial rules ad get a strateg for attackig these problems. The epoetial rules we have leared are: Product Rule: m m + Power to Power Rule: m ( ) ( ) m p m mp p p m p mp Quotiet Rule: m Zero Epoet Rule: Negative Epoet Rule: m 0 Whe simplifig problems that use epoetial rules, ma studets get frustrated because the have o idea where to start or which rule to use. The strateg for attackig problems that use epoetial rules actuall takes us back to the order of operatios. If ou follow the order of operatios, it will guide ou through what ou eed to do. Eample For istace, let's look at the problem that sas: Simplif I order of operatios, the first thig that ou do is the parethesis. Notice that iside the parethesis we ca reduce the fractio. We will first divide 5 ito the 5 ad the 5 to reduce the umbers, the the three s i the deomiator will cacel with three i the umerator to

2 leave a i the top of the fractio, the the three s i the umerator will cacel with three i the deomiator leavig a The result is: 5 i the bottom of the fractio (Notice that for this step we used the Quotiet Rule of subtractig the epoets, sice the fractio bar meas divide.) Now we have performed everthig i the parethesis, so accordig to the order of operatios, the et step is to perform a epoets. There is a square o the outside of the parethesis, so we will eed to square everthig i side the parethesis usig the Power to a Power Rule. This gives us our aswer of: Eample Simplif Let s tr aother eample that looks differet ad see how the order of operatios guides us. ( )( ) ( )( ) ( )( 7 ) Notice that we have a fractio bar that will act as a groupig smbol i the same maer as a parethesis. We must first complete the operatios o the top of the fractio. I the umerator, the first thig we must take care of is the cube o the secod parethesis, which uses the Power to a Power Rule to get: Now we eed to multipl i the umerator sice multiplicatio is doe after epoets. This will require us to use the Product Rule to get:

3 ( )( 7 ) 0 5 Now we ca reduce the fractio, which will require us to use the Quotiet Rule to get: Notice that I did ot take care of the et. I eed to use the Negative Epoet Rule for the. The Negative Epoet Rule requires the to take the reciprocal ad chage the epoet to positive, so sice it is i the bottom of the fractio it will go to the top of the fractio, the be combied with the other usig the Power Rule. This looks like: The aswer is 9. Eample Let s look at aother eample: Simplif 5 7 This problem does ot have a paretheses or a epoets outside paretheses, but it does have some egative epoets that eed to be take care of before we divide. Sice ad 5 both have egative epoets ad the are curretl i the top of the fractio, we will use the Negative Epoet Rule, move them to the bottom ad chage the epoets to positive. Sice the has a egative epoet ad it is i the bottom of the fractio, we will move it to the top of the fractio ad chage its epoet to positive. This will give us:

4 Now we ca use the Product Rule ad multipl the ad 7 to get: Now we ca reduce the umbers ad cacel the two i the umerator with two of the i the deomiator to get: 5 Eample 9 9 The aswer is Oe last eample: 9. Simplif 8 0 We first eed to simplif iside the parethesis if possible. Notice that we ca reduce the umbers ad cacel the s ad the s to get: Now we eed to take care of the epoet. Sice the epoet is egative, we eed to take the reciprocal of the fractio iside the parethesis ad chage the epoet to positive to get: 5 5 Now appl the Power to a Power Rule ad cube everthig to get the aswer of:

5 You are ow read to tr our problems o our homework that ivolve the epoet rules. Keep i mid that it is ver importat to write out ever step! Ma careless mistakes are made from skippig steps i these problems. You are doig great! Keep at it! Whe ou are doe with our problems ivolvig epoet rules come back ad I will show ou a commo applicatio of the epoet rules, scietific otatio.