Official Solution:If \(m\) denotes a number to the left of 0 on the number line such that \(m^{4}\) is less than \(\frac{1}{81}\), then the the reciprocal of \(m^{2}\) must be
A. less than -3
B. between \(-\frac{1}{3}\) and 0
C. between 0 and \(\frac{1}{9}\)
D. between \(\frac{1}{9}\) and 1
E. greater than 9
We must determine which statement correctly the value of, or \(\frac{1}{m^2}\). We are told that \(m \lt 0\) and that \(m^{4} \lt \frac{1}{81}\).
There are two ways to approach this problem. First, it can be solved algebraically. First, take the fourth root of both sides of \(m^{4} \lt \frac{1}{81}\) remembering that since we are looking for the negative root, it is necessary to flip the sign: \(m \gt -\frac{1}{3}\). Next, square both sides, remembering once again to flip the sign: \(m^2 \lt \frac{1}{9}\). Finally, take the reciprocal, remembering that when taking the reciprocal of an inequality in which both sides have the same sign, we must flip the sign. \(m^2\) and \(\frac{1}{9}\) are both positive, so we must flip the sign: \(m^2 \lt \frac{1}{9}\) becomes \(\frac{1}{m^2} \gt 9\). Thus the square of the reciprocal must be greater than 9.
Choice
E is the correct answer.
Alternatively, the problem can be solved by plugging in numbers. We need a negative number which, when raised to the fourth power, is less than \(\frac{1}{81}\). Since fractions get smaller when they have a larger denominator, we can pick a common fourth power greater than \(3^4\) for our denominator like \(4^4 = 256\). Then \(m^4 = \frac{1}{256}\), so \(m = -\frac{1}{4}\). In that case, \(m^2 = \frac{1}{16}\) and the reciprocal of \(m^2\) is 16. Only choice E describes 16.
Answer: E
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