solving for exponent

[joconnor]

Ran into this exponent problem on a practice test and can't find the correct approach. It's more simple than it looks, I'm sure; just not sure of the right first steps:

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

Advice?

[Rahul@gurome]

Ran into this exponent problem on a practice test and can't find the correct approach. It's more simple than it looks, I'm sure; just not sure of the right first steps:

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

Advice?

See the attachment.

[Brian@VeritasPrep]

Hey joconnor:

With exponents, the right first steps are almost always some combination of:

1) Factor out addition/subtraction so that you can multiply divide

2) Find common bases so that you can equate exponents (this usually requires factoring bases into primes)

3) Look for patterns or number properties to make larger numbers easy to deal with

If you have those three "first step" guidelines in your arsenal, you'll almost always be able to get started and the next steps should become clearer.

In this case, the bases of our exponents are 5, 4, and 10. In order to find common bases, we can factor out 4 and 10 into primes:

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

Now, since everything is in the form of multiplication, you have your whole complement of exponent rules available to you [e.g. (x^y)^z = x^yz and (x^a)(x^b) = x^(a+b) ---> all of the exponent rules we have are for multiplication). You can use these rules to expand out the parenthetical terms:

1/5^m * 1/(2^36) = 1 / (2 (2^35 * 5^35))

Then use the common bases to set the exponents equal. Each side has a common 1/2^36, so that divides out, and you're left with 1/5^m = 1/5^35. Therefore, m must be 35.