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Difficulty: HardExponents, Powers, and Square Roots

If aa and bb are nonzero real numbers such that a2b3<0a^2 b^3 < 0 and a2b4=−ab2\sqrt{a^2 b^4} = -a b^2, which of the following statements must be true? Select all such statements.

  1. a+b<0a + b < 0Answer
  2. a3b>0\frac{a^3}{b} > 0Answer
  3. C
    a4b2=a2b\sqrt{a^4 b^2} = a^2 b
  4. D
    (−a)3b2<0(-a)^3 b^2 < 0
  5. E
    (a+b)2=a+b\sqrt{(a + b)^2} = a + b

Answer

The statements that must be true are the inequality asserting that the sum of the variables is negative (a+b<0a + b < 0) and the inequality asserting that the quotient of the cubed variable and the second variable is positive (a3b>0\frac{a^3}{b} > 0).
Analyzing the given constraints reveals that both variables are negative. From a2b3<0a^2 b^3 < 0, since a2>0a^2 > 0, we must have b3<0b^3 < 0, so b<0b < 0. Next, from a2b4=∣a∣b2=−ab2\sqrt{a^2 b^4} = |a| b^2 = -a b^2, dividing by b2>0b^2 > 0 gives ∣a∣=−a|a| = -a, which implies a<0a < 0. Thus, a<0a < 0 and b<0b < 0. The statement asserting a+b<0a + b < 0 is true because the sum of two negative numbers is negative. The statement asserting a3b>0\frac{a^3}{b} > 0 is true because a3<0a^3 < 0 and b<0b < 0, and dividing two negative numbers yields a positive quotient.

Step-by-Step Solution

1
Determine the sign of bb using the given inequality a2b3<0a^2 b^3 < 0.
b<0b < 0
Since aa is a nonzero real number, a2>0a^2 > 0. For the product a2b3a^2 b^3 to be negative, b3b^3 must be negative, which implies b<0b < 0.
2
Determine the sign of aa using the identity a2b4=−ab2\sqrt{a^2 b^4} = -a b^2.
a<0a < 0
Simplify the radical: a2b4=a2⋅(b2)2=∣a∣b2\sqrt{a^2 b^4} = \sqrt{a^2} \cdot \sqrt{(b^2)^2} = |a| b^2. Equating this to −ab2-a b^2 gives ∣a∣b2=−ab2|a| b^2 = -a b^2. Since b≠0b \neq 0, b2>0b^2 > 0, so dividing by b2b^2 yields ∣a∣=−a|a| = -a. For a nonzero real number, ∣a∣=−a|a| = -a implies a<0a < 0.
3
Evaluate statement a+b<0a + b < 0.
True
The sum of two negative numbers (a<0a < 0 and b<0b < 0) is always negative.
4
Evaluate statement a3b>0\frac{a^3}{b} > 0.
True
Since a<0a < 0, a3<0a^3 < 0. Dividing the negative quantity a3a^3 by the negative quantity bb yields a positive result.
5
Evaluate statement a4b2=a2b\sqrt{a^4 b^2} = a^2 b.
False
a4b2=a4b2=a2∣b∣\sqrt{a^4 b^2} = \sqrt{a^4}\sqrt{b^2} = a^2 |b|. Since b<0b < 0, ∣b∣=−b|b| = -b, so a4b2=−a2b\sqrt{a^4 b^2} = -a^2 b.
6
Evaluate statement (−a)3b2<0(-a)^3 b^2 < 0.
False
Since a<0a < 0, −a>0-a > 0, making (−a)3>0(-a)^3 > 0. Since b≠0b \neq 0, b2>0b^2 > 0. The product of two positive numbers is positive, so (−a)3b2>0(-a)^3 b^2 > 0.
7
Evaluate statement (a+b)2=a+b\sqrt{(a + b)^2} = a + b.
False
x2=∣x∣\sqrt{x^2} = |x| for any real xx. Since a+b<0a + b < 0, (a+b)2=∣a+b∣=−(a+b)≠a+b\sqrt{(a + b)^2} = |a + b| = -(a + b) \neq a + b.

Key Concept

Properties of even exponents, odd exponents, and principal square roots of negative variable terms.
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Topics

Exponents, Powers, and Square Roots

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