Surds and Indices
Q. If 2^(x) = 16, what is the value of x?
Solution
16 = 2^(4), so x = 4.
Correct Answer: C — 4
Q. If 2^(x) = 32, what is the value of x?
Solution
32 = 2^(5), so x = 5.
Correct Answer: B — 5
Q. If 3^(x) = 81, what is the value of x?
Solution
81 = 3^(4), so x = 4.
Correct Answer: C — 4
Q. If a = 2^(3) and b = 2^(2), what is the value of a/b?
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A.
2^(1)
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B.
2^(2)
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C.
2^(3)
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D.
2^(5)
Solution
a/b = 2^(3)/2^(2) = 2^(3-2) = 2^(1).
Correct Answer: A — 2^(1)
Q. If a = 3 and b = 2, what is the value of a^(b+1)?
Solution
a^(b+1) = 3^(2+1) = 3^3 = 27.
Correct Answer: C — 27
Q. If a = 3^(1/2), what is a^4?
Solution
a^4 = (3^(1/2))^4 = 3^2 = 9.
Correct Answer: C — 81
Q. If b = 2^(1/3), what is b^6?
Solution
b^6 = (2^(1/3))^6 = 2^2 = 4.
Correct Answer: B — 8
Q. If x = 2^(3) and y = 2^(4), what is the value of x/y?
Solution
x/y = 2^(3)/2^(4) = 2^(3-4) = 2^(-1) = 1/2.
Correct Answer: C — 2
Q. If x = 2^(3/4), what is x^4?
Solution
x^4 = (2^(3/4))^4 = 2^3 = 8.
Correct Answer: C — 8
Q. If x = √(25), what is the value of x^2?
Solution
x = √(25) = 5, thus x^2 = 5^2 = 25.
Correct Answer: C — 25
Q. What is the simplified form of (√(2) * √(8))?
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A.
√(16)
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B.
4
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C.
2√(2)
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D.
8
Solution
√(2) * √(8) = √(16) = 4, but simplified is 2√(2).
Correct Answer: C — 2√(2)
Q. What is the simplified form of (√(a^2) * √(b^2))?
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A.
ab
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B.
a^2b^2
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C.
a + b
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D.
a + b^2
Solution
√(a^2) * √(b^2) = ab.
Correct Answer: A — ab
Q. What is the value of (3^(2))^(3)?
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A.
3^(5)
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B.
3^(6)
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C.
3^(7)
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D.
3^(8)
Solution
Using the property (a^m)^n = a^(m*n), we have (3^(2))^(3) = 3^(2*3) = 3^(6).
Correct Answer: B — 3^(6)
Q. What is the value of (√(2) * √(8))?
Solution
√(2) * √(8) = √(16) = 4.
Correct Answer: B — 4
Q. What is the value of (√(2) + √(3))^2?
Solution
(√(2) + √(3))^2 = 2 + 3 + 2√(6) = 5 + 2√(6).
Correct Answer: C — 7
Q. What is the value of (√(3) + √(2))^2?
Solution
(√(3) + √(2))^2 = 3 + 2 + 2√(6) = 5 + 2√(6).
Correct Answer: C — 10
Q. What is the value of 4^(1/2) * 4^(1/2)?
Solution
4^(1/2) = 2, thus 2 * 2 = 4.
Correct Answer: B — 4
Q. What is the value of 4^(1/2)?
Solution
4^(1/2) = √(4) = 2.
Correct Answer: A — 2
Q. What is the value of 4^(3/2)?
Solution
4^(3/2) = (√4)^3 = 2^3 = 8.
Correct Answer: B — 16
Q. What is the value of 5^(0)?
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A.
0
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B.
1
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C.
5
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D.
undefined
Solution
Any non-zero number raised to the power of 0 is 1.
Correct Answer: B — 1
Q. What is the value of 9^(1/2) + 16^(1/2)?
Solution
9^(1/2) = 3 and 16^(1/2) = 4, so 3 + 4 = 7.
Correct Answer: D — 10
Q. What is the value of √(16) + √(9)?
Solution
√(16) = 4 and √(9) = 3, so 4 + 3 = 7.
Correct Answer: B — 7
Q. What is the value of √(25) - √(16)?
Solution
√(25) = 5 and √(16) = 4, so 5 - 4 = 1.
Correct Answer: B — 3
Q. Which of the following is equal to 5^(2/3)?
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A.
√(25)
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B.
5√(5)
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C.
25
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D.
√(125)
Solution
5^(2/3) = (5^(1/3))^2 = 5√(5).
Correct Answer: B — 5√(5)
Q. Which of the following is equivalent to (x^(1/2))^4?
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A.
x^(2)
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B.
x^(1)
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C.
x^(1/4)
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D.
x^(1/8)
Solution
Using the power of a power property, (x^(1/2))^4 = x^(1/2 * 4) = x^(2).
Correct Answer: A — x^(2)
Q. Which of the following is equivalent to 2^(3) * 2^(2)?
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A.
2^(5)
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B.
2^(6)
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C.
2^(7)
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D.
2^(8)
Solution
Using the property a^m * a^n = a^(m+n), we have 2^(3+2) = 2^(5).
Correct Answer: A — 2^(5)
Q. Which of the following is equivalent to 2^(3/2)?
Solution
2^(3/2) = √(2^3) = √8.
Correct Answer: A — √8
Q. Which of the following is true for the expression 3^(2x) = 27?
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A.
x = 1
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B.
x = 2
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C.
x = 3
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D.
x = 0
Solution
27 = 3^(3), so 2x = 3, thus x = 3/2.
Correct Answer: A — x = 1
Q. Which of the following is true for the expression 4^(x) = 16?
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A.
x = 1
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B.
x = 2
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C.
x = 3
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D.
x = 4
Solution
16 = 4^(2), thus x = 2.
Correct Answer: B — x = 2
Q. Which of the following is true for the expression 5^(x) = 25?
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A.
x = 1
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B.
x = 2
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C.
x = 3
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D.
x = 4
Solution
25 = 5^(2), thus x = 2.
Correct Answer: B — x = 2
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