Solutions

  1. Remember that a negative exponent yields the reciprocal of the same expression with a positive exponent. Thus, .

  2. 49

  3. 404

    84(54) = 404

  4. −12

  5. 4,177

  6. No

    Remember, you cannot combine exponential expressions linked by addition.

  7. Any non-zero number less than 1

    As positive proper fractions are multiplied, their value decreases. For example, . Also, any negative number will make this inequality true. A negative number cubed is negative. Any negative number squared is positive. For example, (−3)3 < (−3)2. The number zero itself, however, does not work, because 03 = 02.

    This could be determined algebraically:


    x    2 is positive for all x ≠ 0, so x2(x − 1) is negative when (x − 1) is negative: x < 1.

  8. 2

    The possible values for x are 2 and −2. The absolute value of both 2 and −2 is 2.

  9. No

    An integer raised to an odd exponent retains the original sign of the base. Therefore, if y5 is positive, y is positive.

  10. Yes

    b and a are both positive numbers. Whether c is positive or negative, c4 is positive. (Recall that any number raised to an even power is positive.) Therefore, the product a2b3c4 is the product of three positive numbers, which will be positive.

  12. 0, −1

    If r3 + | r | = 0, then r3 must be the opposite of | r |. The only values for which this would be true are 0, which is the opposite of itself, and −1, whose opposite is 1.

  13. (C)

    When you raise a number to a negative power, that’s the same as raising its reciprocal to the positive version of that power. For instance, , because is the reciprocal of 3. The reciprocal of is 2, so Quantity B can be rewritten.

    Quantity A Quantity B
    2y

    Therefore, the two quantities are equal.

  14. (A)

    The goal with exponent questions is always to get the same bases, the simplest versions of which will always be prime. Each quantity has the same four bases: 2, 3, 4, and 9. Because 2 and 3 are already prime, you need to manipulate 4 and 9: 4 = 22 and 9 = 32. Rewrite the quantities:

    Quantity A Quantity B
    33 × 96 × 24 × 42 =
    33 × (32)6 × 24 × (22)2    		 93 × 36 × 22 × 44 =
    (32)3 × 36 × 22 × (22)4

    Now terms can be combined using the exponent rules.

    Quantity A Quantity B
    33 × (32)6 × 24 × (22)2 =
    33 × 312 × 24 × 24 =
    315 × 28    		 (32)3 × 36 × 22 × (22)4 =
    36 × 36 × 22 × 28 =
    312 × 210

    Now divide away common terms. Both quantities contain the product 312 · 28.

    Quantity A Quantity B

    Therefore, Quantity A is greater.

  15. (B)

    Any number less than 1 raised to a power greater than 1 will get smaller, so even though you don’t know the value of y, you do know that the value in Quantity A will be less than 0.99:

              y > 1

    Quantity A Quantity B
    (0.99)y, which must be less than 0.99 0.99 × y

    Conversely, any positive number multiplied by a number greater than 1 will get bigger. You don’t know the value in Quantity B, but you know that it will be larger than 0.99:

    y > 1

    Quantity A Quantity B
    (0.99)y, which must be less than 0.99 0.99 × y, which must be greater than 0.99

    Therefore, Quantity B is greater.