Exponent question

The question is incorrect

If 3x - 3x-1 = 162, this means -1 = 162.

Can you pls correct it

I love the question and rerika's attempt to post the exponents as actual exponents (rather than the x^y) form, but it didn't quite translate.

I think I can break down the question, though (and it fits with the answer choices). It should read:


If 3^x - 3^(x-1) = 162, then what is x(x-1)?

A) 12
B) 16
C) 20
D) 30
E) 81


As it's written, the answer is C.





Explanation (I'll put in "spoiler" text since many have not been able to read the question yet):

[spoiler]It may seem awkward to do, but you can factor out a 3^(x-1) from both terms on the left hand side. 3^x can be written as 3^1 * 3^(x-1) in order to create that duplicate term to factor:

3^x - 3^(x-1) = 162
3^1 * 3^(x-1) - 3^(x-1) = 162
[3^(x-1)] (3-1) = 162
[3^(x-1)] * 2 = 162


Then you can divide both sides by 2 to get:

3^(x-1) = 81

As with any exponent problem with variables in the exponents, you should try to create the same bases:

3^(x-1) = 3^4

x - 1 = 4
x = 5

x(x-1) = 5*4 = 20

That's one, fairly direct way to do it. Another way would be to recognize that you need to find 3s on the right hand side of the equation to be able to do anything with the 3^variable terms on the left, and factor 162 into 2*3^4. Then you could either backsolve combinations of x and x-1 or work to find patterns with 3s and consecutive exponents.

Ultimately, I'd say that the biggest takeaway on this problem is that, when exponents appear with variables in the exponents, you should try to find common bases, and usually to do that you'll want to break numbers down into prime factors. [/spoiler]

rerika01 wrote:I am trying to figure out how to solve this question. Thanks


If 3^x - 3^x-1 =162, then x(x-1)=?
A) 12
B) 16
C) 20
D) 30
E) 81

3^x - 3^x-1 =162

factorise 162....
Right hand Side of the equation : 162 = 3^4 * 2
[spoiler]left Hand side of the equation :
3^x - 3^x-1 = 3^x(1 - 1/3) = 3^x * (2/3) = 3^(x-1) * 2

So,
3^(x-1) * 2 = 3^4 * 2
So (x-1) = 4 which implies x = 5
So (x-1)x = 4*5 = 20[/spoiler]

Hey Brian,

I was actually thinking of this problem this way:

3^x - 3^(x-1) =162
3^x - 3^(x-1) = 3^4 + 3^4

Now I have the same bases, could I just take the exponents from here on?

x-(x-1)=4+4 <-- would this be right?

Thanks,

Hey Gurpinder,

I like the thought, but as you've probably noticed you end up with a nonsense equation from there:

x - (x-1) = 4+4
1 = 8 ----> Which is obviously not correct, and the xs are long gone.

The reason for that is that we can only strip off the bases if we have a one-for-one equality (like 3^(x-1) = 3^4). When addition or subtraction of exponents is present, we just don't have enough to be able to break off the exponents (a pretty important concept...we're really good at multiplication/division of exponents but terrible at adding/subtracting).

Think of it this way - we only add exponents when we're multiplying their bases. Here, we're adding/subtracting their bases, so we don't have any indication that we can combine the exponents. 3^4 + 3^4 is not 3^8...

Remember this - when using exponents, the majority of the rules we have are for multiplication/division of the same bases, so your goal should be to use factoring to get you in a position to use those rules.

Brian@VeritasPrep wrote:Hey Gurpinder,

I like the thought, but as you've probably noticed you end up with a nonsense equation from there:

x - (x-1) = 4+4
1 = 8 ----> Which is obviously not correct, and the xs are long gone.

The reason for that is that we can only strip off the bases if we have a one-for-one equality (like 3^(x-1) = 3^4). When addition or subtraction of exponents is present, we just don't have enough to be able to break off the exponents (a pretty important concept...we're really good at multiplication/division of exponents but terrible at adding/subtracting).

Think of it this way - we only add exponents when we're multiplying their bases. Here, we're adding/subtracting their bases, so we don't have any indication that we can combine the exponents. 3^4 + 3^4 is not 3^8...

Remember this - when using exponents, the majority of the rules we have are for multiplication/division of the same bases, so your goal should be to use factoring to get you in a position to use those rules.

I kind of had the feeling that I just got a meaningless equation right in the beginning!

GREAT Tips!

Thanks Brian.

rerika01 wrote:I am trying to figure out how to solve this question. Thanks


If 3^x - 3^x-1 =162, then x(x-1)=?
A) 12
B) 16
C) 20
D) 30
E) 81

3^x -3^(x-1)=3^(x)-3^(x)/3=2*3^4 or 3*3^x-3^x=2*3^5 or 2*3^x=2*3^5
thus x= 5
and 5(5-1)=20
the answer is C

Good tips on using exponents...it's helpful to think of it that way that you should use factoring to get to a point where you can use the multiplication rules for exponents. Now that I think of it I've seen a lot of multiplication with exponents in practice but the addition/subtraction is harder, I guess because you have to turn it into multiplication to get it to work. Thanks!

Hi,

I did it as below:

3^x can be grouped as a unit as say y.

So the equation now becomes :

y - y/3 = 162.

Solving this gets me y = 243.
So y = 3^x = 243 which solves as to x =5.

x(x-1) = 5 X 4 = 20 (C)

Here is how I solved this:

3^x - 3^(x-1) = 162
3^x -(3^x)(3^-1)= 162
3^x(1-1/3)= 162
3^x(2/3)=162
3^x= 162 * 3/2
3^x = 81 * 3
x = 5

5 * (5-1) = 20

C. 20