if the value of \frac{4^m}{4^n}<1, which of the following must be true
A. m=1
B. n=1
C. m=n
D. m<n
E. m>n
4^m/4^n < 1
4^(m-n) < 1
you know that 4^0 = 1
so, what do you think? which powers of 4 are less than 1?
im confused
To determine which of the given options must be true, let's first simplify the expression \(\frac{4^m}{4^n}\).
Using the properties of exponents, we know that \(a^m / a^n = a^{m-n}\). Applying this to our expression, we have \(\frac{4^m}{4^n} = 4^{m-n}\).
Since the value of \(\frac{4^m}{4^n}\) is less than 1, we can conclude that \(4^{m-n}\) must also be less than 1.
Recall that \(a^0 = 1\) for any non-zero number a. So, if \(m-n = 0\), then \(4^{m-n}\) equals \(4^0\), which is equal to 1. This means that \(m\) cannot be equal to \(n\) (option C cannot be true).
Similarly, if \(m-n > 0\), then \(4^{m-n}\) would be a positive number greater than 1. Therefore, \(4^{m-n}\) cannot be less than 1 when \(m > n\). Hence, option D (m < n) cannot be true.
Now, if \(m-n < 0\), then \(4^{m-n}\) would be a positive number greater than 1, i.e., \(4^{m-n} > 1\). So, \(4^m / 4^n\) can only be less than 1 when \(m-n\) is either 0 or negative.
From this analysis, we conclude that the only valid answer is option E: \(m > n\).