1. 3⁴ The number of repeated multiplications of the base by itself
81
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: The exponent will always denote the repeated multiplication of a number by itself; for example, 3⁴ = (3)(3)(3)(3) = 4 repeated multiplications 3⁴ = (3)(3)(3)(3) = 81 81 > 4
2. 3⁴ + (12! - 100)
(5)(5)(5)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Let’s separate the left side into pieces; 3⁴ = (3)(3)(3)(3) = 81 and (12! - 100) = 144 - 100 = 44 81 + 44 = 125 And, for the right side; (5)(5)(5) = (25)(5) = 125 125 = 125
3. 3⁴ The base divided by The exponent divided the exponent by the base
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: In the expression 3⁴, 3 is called the base, 4 is called the exponent, and we read the expression as “3 to the fourth power.” The base divided by the exponent = 3/4 The exponent divided by the base = 4/3 4/3 > 3/4
4.
5 to the third power
125
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: 5 to the third power is 5" = 125. When the exponent is 2, we call the process squaring. Thus, 6 squared is 36, 6! = (6)(6) = 36, and 7 squared is 49, 7! = (7)(7) = 49.
5. (-3)!
(-3)â ľ
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: When negative numbers are raised to powers, the result may be positive or negative. For example, (-3)! = (-3)(-3) = 9, while (-3)â ľ = (-3)(-3)(-3) (-3)(-3) = -243. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative.
6. (-3)"
(-3)â ľ
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: When negative numbers are raised to odd powers, the result will be negative. For example, (-3)" = (-3)(-3)(-3) = -27, while (-3)â ľ = (-3)(-3) (-3)(-3)(-3) = -243. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. -27 > -243
7. (-3)!
(-3)â ´
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: When negative numbers are raised to even powers, the result will be positive. For example, (-3)! = (-3)(-3) = 9, while (-3)â ´ = (-3)(-3)(-3)(-3) = 81. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. 81 > 9
8. -3!
-(3)!
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Without the parentheses, the expression -3! means “the negative of ‘3 squared’ ”; that is, the exponent is applied before the negative sign. So (-3)! = 9, but -3! = -9.
9. 3â °
(3)â °
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation:
Exponents can also be negative or zero; such exponents are defined as follows. For all nonzero numbers a, a⁰ = 1. For example, 3⁰=1. (3)⁰ is also 1.
10. Which of the following is FALSE? A. Exponents are used to denote the repeated multiplication of a number by itself B. When negative numbers are raised to powers, the result may be positive or negative C. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative. D. The expression 0⁰ is 1. E. A square root of a nonnegative number n is a number r such that r! = n. Correct: D Explanation: D. The expression 0⁰ is undefined.
A square root of a nonnegative number n is a number r such that r! = n. A square root of a nonnegative number 16 is a number 4 such that 4! = 16. A square root of a nonnegative number 25 is a number 5 such that 5! = 25. In other words, 4 is a square root of 16 because 4! = 16.
11.
3â ť#
3â ť!
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A
Explanation: 3⁻# = 1/3# = 1/3 3⁻! = 1/3! = 1/9 a⁻# = 1/a# = 1/a a⁻! = 1/a! a⁻" = 1/a" 1/3 > 1/9
12. Assume a $ 0
a⁰
(a)(a⁻#)
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information
Correct: C Explanation: a⁰ = 1 (a)(a⁻#) = (a)(1/a) = 1
13.
4
The number of square roots of 16
A.The quantity on the left is greater B. The quantity on the right is greater
C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: 4 is a square root of 16 because 4! = 16. Another square root of 16 is -4, since (-4)! = 16. There are two square roots of 16, 4 and -4. Therefore, the left is greater because 4 is greater than 2.
14. The number of square The number of square roots that all positive roots that zero has numbers have
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: All positive numbers have two square roots, one positive and one negative. The only square root of 0 is 0. Therefore, the left is greater than the right because 2 is greater than 1.
15.
-%100
%100
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: The symbol % is used to denote the nonnegative square root of the nonnegative number n. Therefore, %100 = 10 and only 10. - %100 = -10
16.
%0
-3â ť!
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: The square root of 0 is 0. -3â ť! = - 1/3! = - 1/9 0 > -1/9
17.
(%a)!
%a!
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: The exponent rule is clear; (%a)! = a %a! = a
18.
(%b)(%a)
%ab
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: The exponent rule is clear; (%b)(%a) = %ab For example, (%2)(%3) = %6
19.
(%b)/(%a)
%b/a
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: The exponent rule is clear; (%b)/(%a) = %b/a For example, (%6)/(%2) = %3 %6/2 = %3
20.
(%3)!
%4
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: A Explanation: Like question 17, The exponent rule is clear; (%3)!= 3 %4 = 2 3>2
21.
%3%10
%30
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: Like question 18, The exponent rule is clear; (%b)(%a) = %ab %3%10 = %(3)(10) = %30
22.
(%5)/(%15)
%3/1
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation: The exponent rule is clear; (%b)/(%a) = %b/a For example, (%5)/(%15) = %5/15 = %1/3 %3/1 > %1/3
23.
(%&)!
%&!
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: The exponent rule is clear; (%&)! = & %&! = &
24.
%24
2%6
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: %24 = %4%6 = 2%6
25.
%18/%2
3
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation: %18/%2 = %18/2 = %9 = 3
26.
∛8
2
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: C Explanation:
A square root is a root of order 2. For orders 3 and 4, we say the cube root ∛ and fourth root ∜ represent numbers such that when they are raised to the powers 3 and 4, respectively, the result is n. ∛8 = 2 ∛27 = 3 ∛64 = 4 ∛125 = 5 ∜16 = 2 ∜81 = 3 ∜254 = 4 These roots obey rules similar to those above in questions 17, 18 and 19. (but with the exponent 2 replaced by 3 or 4 in the first two rules). There are some notable differences between odd- order roots and evenorder roots (in the real number system): For odd-order roots, there is exactly one root for every number n, even when n is negative. For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n.
27. The number of cube The number of fourth roots of 8 roots of 8
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information Correct: B Explanation:
There are some notable differences between odd- order roots and evenorder roots (in the real number system): For odd-order roots, there is exactly one root for every number n, even when n is negative. For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n. The right is greater than the left because 2 is greater than 1.
28. The number of cube The number of fourth roots of -8 roots of -8
A.The quantity on the left is greater B. The quantity on the right is greater C. Both are equal D. The relationship cannot be determined without further information
Correct: A Explanation: There are some notable differences between odd- order roots and evenorder roots (in the real number system): For odd-order roots, there is exactly one root for every number n, even when n is negative. For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n. The left is greater than the right because 1 is greater than 0.