If [(0.0015)(10^m)]/[(0.03)(10^k)] = 5(10^7)…
We start by simplifying the numerator and denominator of the given fraction. First, we simplify the numerator:
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:
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)
Dividing 15 by 3 on the left hand side of the equation, we have 15/3 = 5. Recall that when we divide powers of like bases, we subtract the exponents, so 10^(m-4)/10^(k-2) =
10^((m-4) – (k-2)) = 10^(m-k-2). Therefore, we have
5(10^(m-k-2)) = (5)(10^7)
5 will cancel out from both sides of the equation, leaving us with:
Because we are left with a base of 10 on both the right-hand side and the left-hand side of the equation, we can drop the base and set the exponents equal and hence determine the value of m – k:
m – k – 2 = 7
m – k = 9