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Quantity A is greater. Quantity B is greater. The two quantities are equal. The relationship cannot be determined from the information given.
GIVEN: \(2^p=4^q\) Rewrite 4 as 2² to get: \(2^p=(2^2)^q\) Apply Power of a Power law to get: \(2^p=2^{2q}\) Since we have the same base, we can conclude that \(p=2q\)
Can we use values for p and q here ( say, p=1, So p>q or p=0, so , p=q) to determine their relationship?
Yes, you can do that. The only downside with that strategy (testing values) is that you can never be 100% sure of the correct answer UNLESS you find two conflicting cases (which means the correct answer is D).
For example, if 2^p + 4^q, then one possibility is that p = 0 and q = 0 In this case, p = 2q
Another possibility is that p = 2 and q = 1 In this case, p = 2q
Another possibility is that p = 4 and q = 2 In this case, p = 2q
Another possibility is that p = 6 and q = 3 In this case, p = 2q
At this point, it certainly SEEMS like the correct answer is C. But, perhaps there's a pair of values that we haven't checked yet and those values would show that p ≠ 2q
Although the correct answer is, indeed, C, we can't be 100% sure of that when we test values.
I discuss those limitations in the following video (starting at 2:50)