Math Problem (supposedly high school level)

Dibs

This is a question from a GMAT practice exam -- I have no idea how they got the answer, and I'm hoping someone here can help me. I tried an online solver and it didn't give me the correct answer, but I know there's some real math geniuses here so I thought I'd give it a shot.

Here is the problem:

(1/4)^m * (1/5)^18 = 1/2(10)^35
m = ?

Thoughts? I have asked everyone I know for help here, and we are all drawing a blank.

The correct answer is 35... seems to me (and everyone else) like it should be 17? Is there some rule I'm missing?

Hirocon

Are you sure you've written the problem correctly? As you've written it, the answer won't be an integer. I can't tell if you meant for the 10 to be in the numerator or the denominator of the fraction on the right, but either way the answer won't be an integer.

PlushyCthulhu

Is this from a test book? Some of them are horribly, horribly written and in all likelihood the question is just screwed up and/or the "correct" answer is flat out wrong I'm guessing the intended equation was something like (1/4)^17 * (1/5) ^ 35 = 2 * (1/10) ^ 35

KafkaAU

There is something not written correctly there. Neither 17 or 35 is correct as to how you have written it, it would be something quite negative. The number on the left is really small and the number on the right is really large.

Is the equation how PlushyCthulhu wrote it (with the 1/5 power corrected to 18? i.e. (1/4)^m * (1/5)^18 = 2 * (1/10)^35

In any case you solve it using logarithms.

Mushiwulf

It is written incorrectly here. Swap 1/4 and 1/5. Then solve.

Dibs

1) Sorry being unresponsive.

2) I thought I double checked it, but apparently I still put it in wrong. Here is an image as it is displayed in the practise exam. Mushi was correct, the 1/4 and 1/5 are swapped (but contrary to Cthulu, the 2 is definitely only on the bottom.. or at least that's my understanding).



You can see my guess (solid circle) and the correct answer (squared answer). I am still lost on this.

Sorry for wasting people's time, I feel like an idiot for mistyping it. I knew it was going to be a challenge to get a math problem across properly, but I'm surprised I messed up the simple part.

Hahnsoo1

Good lord, that changes everything.

So, convert (1/4)^18 to 1/(2)^36 (because 4 = 2^2, and (2^2)^18 converts to 2^36). Then multiply both sides of the equation by 2^36. The right side then becomes 2^36/(2(10)^35), which then becomes 2^35/10^35*, which then becomes, (2/10)^35. Thus (1/5)^m = (2/10)^35. 2/10 = 1/5, leaving you with your answer.

* 2^36 divided by 2 is 2^35. Or another way to put it is 2^36/2^1 = 2^(36-1) = 2^35

tarnok

To put it another way;

(1/4)^18 * (1/5)^m = 1/(2*10^35) :given
[(1/2)^2]^18 * (1/5)^m = 1/(2*10^35) : because (1/2)^2 = 1/4
(1/2)^36 * (1/5)^m = 1/(2*10^35) : power rule for exponents
(1/2)^36 * (1/5)^m = 1/(2 * [2^35 * 5^35]) : product rule for exponents
(1/2)^36 * (1/5)^m = 1/(2^36 * 5^35) : multiplying powers of the same base
(1/2)^36 * (1/5)^m = 1/(2^36) * 1/(5^35) : definition of multiplication for fractions
(1^36)/(2^36) * (1^m)/(5^m) = 1/(2^36) * 1/(5^35) : quotient rule for exponents
1/(2^36) * 1/(5^m) = 1/(2^36) * 1/(5^35) : 1^x = 1 for any x != 0
Since both sides are identical except for the variable the variable must be equal to the number occupying its place on the right side.

shwaip

also if you don't like fractions, you can flip both sides:

5^m * 4^18 = 2*10^35
5^m * 4^18 = 2 * 2^35 * 5^35
5^m * 2^36 = 2^36 * 5^35
5^m = 5^35
m = 35