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Difficulty: Very hardExponents, Powers, and Square Roots

If xx and yy are real numbers such that −1<x<0<y<1-1 < x < 0 < y < 1, which of the following statements MUST be true? Select all such statements.

  1. x3<x5x^3 < x^5Answer
  2. B
    x2=x\sqrt{x^2} = x
  3. (y1/2)x>1\left(y^{1/2}\right)^x > 1Answer
  4. D
    (x+y)2=x2+y2(x + y)^2 = x^2 + y^2
  5. E
    x4>x2x^4 > x^2

Answer

The statements x3<x5x^3 < x^5 and (y1/2)x>1\left(y^{1/2}\right)^x > 1 MUST be true.
For −1<x<0<y<1-1 < x < 0 < y < 1, odd powers of xx satisfy −1<x<x3<x5<0-1 < x < x^3 < x^5 < 0, making the inequality comparing x3x^3 and x5x^5 true. Furthermore, y1/2y^{1/2} lies in (0,1)(0, 1), and raising a base in (0,1)(0, 1) to a negative exponent xx produces a result strictly greater than 1.

Step-by-Step Solution

1
Analyze the odd power inequality x3<x5x^3 < x^5 for −1<x<0-1 < x < 0.
Since x∈(−1,0)x \in (-1, 0), x2∈(0,1)x^2 \in (0, 1). Multiplying −1<x<0-1 < x < 0 by x2x^2 gives x<x3<x5<0x < x^3 < x^5 < 0. Thus, x3<x5x^3 < x^5 holds.
Odd powers preserve negative signs, and higher powers of fractions between 0 and 1 have smaller absolute values.
2
Evaluate the principal square root x2\sqrt{x^2}.
By definition, x2=∣x∣\sqrt{x^2} = |x|. Since x<0x < 0, ∣x∣=−x≠x|x| = -x \neq x.
The principal square root of a real number is always non-negative.
3
Analyze the expression (y1/2)x\left(y^{1/2}\right)^x.
Since 0<y<10 < y < 1, 0<y1/2<10 < y^{1/2} < 1. Let k=y1/2k = y^{1/2}. Then kx=(1k)−xk^x = \left(\frac{1}{k}\right)^{-x}. Since k<1k < 1, 1k>1\frac{1}{k} > 1, and since x<0x < 0, −x>0-x > 0. A base greater than 1 raised to a positive power is greater than 1.
Negative exponents indicate the reciprocal of the base.
4
Evaluate (x+y)2=x2+y2(x + y)^2 = x^2 + y^2.
(x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2. Because x<0x < 0 and y>0y > 0, 2xy<02xy < 0, so (x+y)2<x2+y2(x + y)^2 < x^2 + y^2.
Exponents do not distribute over addition, and cross-terms must be accounted for.
5
Evaluate x4>x2x^4 > x^2.
For −1<x<0-1 < x < 0, 0<x2<10 < x^2 < 1. Squaring a number in (0,1)(0, 1) yields a smaller number, so x4<x2x^4 < x^2.
Higher even powers of quantities with magnitude less than 1 decrease in magnitude.

Key Concept

Properties of exponents, fractional powers, and principal square roots for bounded negative and positive real numbers
Estimated Time:2m 30s
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Topics

Exponents, Powers, and Square Roots

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