Lots of exponents

JJMforthegold wrote:Experts-

Is there a fast way solve this without multiplying everything out?

(0.0015 x 10^m)/(0.03 x 10^k) = 5 x 10^7, then m - k =

a) 9
b) 8
c) 7
d) 6
e) 5

Let k=0.
(0.0015 * 10^m) / (0.03* 10^0) = (5 * 10^7)

Convert the decimals to integers:
(15 * 10^(-4) * 10^m / (3 * 10^(-2) * 1) = (5 * 10^7)

Divide the integers and subtract the exponents:
5 * 10^(-2) * 10^m = 5 * 10^7

Cancel the 5's and divide each side by 10^(-2):
10^m = 10^9
m = 9.

Thus:
m-k = 9-0 = 9.

The correct answer is A.

JJMforthegold wrote:Experts-

Is there a fast way solve this without multiplying everything out?

(0.0015 x 10^m)/(0.03 x 10^k) = 5 x 10^7, then m - k =

a) 9
b) 8
c) 7
d) 6
e) 5

We start by simplifying the numerator and denominator of the given fraction. First, we simplify the numerator:

(0.0015)(10^m)

It will be helpful to convert 0.0015 to an integer. To do so we must move the decimal point in 0.0015 four places to the right. Since we are making 0.0015 larger by four decimal places we must make 10^m, smaller by four decimal places. Thus, 10^m now becomes 10^(m-4). Thus, the numerator becomes (15)(10^(m-4)).

Next we can simplify the denominator:

(0.03)(10^k)

It will be helpful to convert 0.03 to an integer. To do so we must move the decimal point in 0.03 two places to the right. Since we are making 0.03 larger by two decimal places we must make 10^k, smaller by two decimal places. Thus, 10^k now becomes 10^(k-2). The denominator can thus be re-expressed as (3)(10^(k-2)).

So now we are left with:

[(15)(10^(m-4))]/[(3)(10^(k-2))] = 5(10^7)

5(10^(m-4 - (k-2))) = (5)(10^7)

10^(m-k-2)=10^7

m – k – 2 = 7

m – k = 9

Answer: A