I'm supposed to be working on the laws of exponents. answers are supposed to be in form ae^b
\[\sqrt[3]{-8e ^{6x}} \] and \[(e ^{2x}e ^{3x})/e ^{5}\]
For the first one, recall that \[\huge \sqrt[a]{b}=b^{\frac{1}{a}}\]
so with that just to make sure..would i pull the negative along when i pull out the 8?
\[8^{ \frac{ 6x }{ 3}}\]
Keep it there. It's the safest thing to do... Now... \[\huge \sqrt[3]{-8e^{6x}}=\left(-8e^{6x}\right)^{\frac{1}{3}}\]
so then i would have 6x/3 for my exponent? this is the point where i usually start getting confused because of all the rules with exponents
It's okay, that's why you have me :) We're taking things one at a time, here... So, remember this law of exponents \[\huge (ab)^m=a^mb^m\] ? Now would be a good time to apply it \[\huge \left(-8e^{6x}\right)^{\frac{1}{3}}=\left[\left(-8\right)\left(e^{6x}\right)\right]^{\frac{1}{3}}\]
okay so then i would have (2) and the other part will be e^6/3x ?
i mean -2
Very good :) I was about to correct you, but you beat me to it. That's why I wanted to keep the negative sign :) So, yeah, you get \[\huge -2e^{\frac{6x}{3}}\] And now just simplify that fraction. Before anything else, that fractional exponent is a fraction, which can be reduced to lowest terms:)
so then i would have \[-2e ^{2x}\]
Precisely. Well done there :)
wow. thank you so much!! I would not have figured that out on my own!
In time, you'll get the hang of it, and it'll all be second nature :) Now, you need help with the second question?
yes i do.
i know with exponents and division there is some kind of subtraction, but I couldn't get it
Well, this question is, in my opinion, simpler than the first one~ It doesn't involve radical signs :D Just remember these two rules. \[\huge a^ma^n=a^{m+n}\] \[\huge \frac{a^m}{a^n}=a^{m-n}=\frac{1}{a^{n-m}}\]
So, let's begin :) You're given \[\huge \frac{e^{2x}e^{3x}}{e^{5}}\] Let's do things one at a time, here...
In the numerator, you multiply two exponentials with the same base. Using the rule posted above, you can just add their exponents, right? Go ahead now...
e^5x
That's right, so you're left with \[\huge \frac{e^{5x}}{e^5}\] Careful here, there are so many ways to go wrong... Use the rule about division.
1/e^x or would it be 1/e^5x-5 ?
Neither. Now review this rule again. I've only put in once instance so you don't get confused :) \[\huge \frac{a^m}{a^n}=a^{m-n}\]
e^5x-5 i would keep the e right? because that kind of represents the a and since the 5x and 5 are not the same i couldn't directly subtract them to get an actual number
Yep. You're left with \[\huge e^{5x-5}\] I don't see a way to further simplify this...maybe \[\huge e^{5(x-1)}\] But in any case, good job :)
yeah! thank you so much!! I really need to go look online and find all those exponent rules again. thank you so much for your help
No problem :)
Join our real-time social learning platform and learn together with your friends!