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Difficulty: MediumExponents, Powers, and Square Roots

If mm and nn are negative integers such that m<nm < n, which of the following statements must be true? Select all such statements.

  1. 2m<2n2^m < 2^nAnswer
  2. (12)m>(12)n\left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^nAnswer
  3. C
    m2<n2m^2 < n^2
  4. D
    m2=m\sqrt{m^2} = m
  5. E
    2m+n=2m+2n2^{m+n} = 2^m + 2^n

Answer

The statements 2m<2n2^m < 2^n and \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n must be true.
For the statement involving base 22, since 2>12 > 1, the exponential function is strictly increasing, so m<nm < n guarantees 2m<2n2^m < 2^n. For the statement involving base 12\frac{1}{2}, since 0<12<10 < \frac{1}{2} < 1, the function is strictly decreasing, meaning a smaller input mm produces a larger output, so \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n.

Step-by-Step Solution

1
Analyze the expression 2m<2n2^m < 2^n for base greater than 1
Since b=2>1b = 2 > 1, raising 22 to a larger exponent yields a larger value. Because m<nm < n, 2m<2n2^m < 2^n is always true.
Exponential functions with a base b>1b > 1 are strictly increasing.
2
Analyze the expression \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n for fractional base between 0 and 1
Since b=12b = \frac{1}{2} is between 00 and 11, raising 12\frac{1}{2} to a smaller exponent yields a larger value. Because m<nm < n, \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n is always true.
Exponential functions with a base 0<b<10 < b < 1 are strictly decreasing.
3
Evaluate the remaining algebraic statements using counterexamples
For m=−3m = -3 and n=−2n = -2: m2=9>4=n2m^2 = 9 > 4 = n^2, so m2<n2m^2 < n^2 is false. m2=9=3≠−3\sqrt{m^2} = \sqrt{9} = 3 \neq -3, so m2=m\sqrt{m^2} = m is false. 2−5=132≠18+14=382^{-5} = \frac{1}{32} \neq \frac{1}{8} + \frac{1}{4} = \frac{3}{8}, so 2m+n=2m+2n2^{m+n} = 2^m + 2^n is false.
A single counterexample disproves that a statement MUST be true.

Key Concept

Monotonicity of exponential functions and properties of square roots of negative bases
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Topics

Exponents, Powers, and Square Roots

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