Q. If 10^(x) = 1000, what is the value of x? (2023)
Solution
Since 1000 can be expressed as 10^3, we have 10^x = 10^3, thus x = 3.
Correct Answer:
B
— 3
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Q. If 10^(x+1) = 1000, what is the value of x?
Solution
1000 can be expressed as 10^3, so x + 1 = 3, leading to x = 2.
Correct Answer:
B
— 2
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Q. If 10^(x+2) = 1000, what is the value of x? (2023)
Solution
Since 1000 can be expressed as 10^3, we have 10^(x+2) = 10^3, thus x + 2 = 3, leading to x = 1.
Correct Answer:
A
— 1
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Q. If 2^(x+3) = 32, what is the value of x?
Solution
Since 32 can be expressed as 2^5, we have 2^(x+3) = 2^5, thus x + 3 = 5, leading to x = 2.
Correct Answer:
C
— 3
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Q. If 4^(x-1) = 1/16, what is the value of x? (2023)
Solution
Since 1/16 can be expressed as 4^(-2), we have 4^(x-1) = 4^(-2), thus x - 1 = -2, leading to x = -1.
Correct Answer:
C
— 2
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Q. If 4^(x-1) = 64, what is the value of x?
Solution
Since 64 can be expressed as 4^3, we have 4^(x-1) = 4^3, thus x - 1 = 3, leading to x = 4.
Correct Answer:
B
— 4
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Q. If 7^(2x) = 49, what is the value of x? (2023)
Solution
Since 49 can be expressed as 7^2, we have 7^(2x) = 7^2, thus 2x = 2, leading to x = 1.
Correct Answer:
B
— 1
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Q. If 7^(x) = 1/49, what is the value of x? (2023)
Solution
Since 1/49 can be expressed as 7^(-2), we have 7^x = 7^(-2), thus x = -2.
Correct Answer:
A
— -2
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Q. If a = 2 and b = 3, what is the value of a^b + b^a?
Solution
Calculating, a^b = 2^3 = 8 and b^a = 3^2 = 9, thus a^b + b^a = 8 + 9 = 17.
Correct Answer:
B
— 17
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Q. If a = 3 and b = 2, what is the value of a^b + b^a?
Solution
Calculating 3^2 = 9 and 2^3 = 8, thus 9 + 8 = 17.
Correct Answer:
B
— 17
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Q. If a^0 = 1 for any non-zero number a, what can be inferred about the expression 5^0?
-
A.
It equals 0.
-
B.
It equals 1.
-
C.
It is undefined.
-
D.
It equals 5.
Solution
According to the exponent rule, any non-zero number raised to the power of zero equals 1.
Correct Answer:
B
— It equals 1.
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Q. If a^0 = 1 for any non-zero number a, which of the following is true?
-
A.
0^0 is also equal to 1.
-
B.
1^0 is equal to 0.
-
C.
Any number raised to the power of 0 is undefined.
-
D.
Only positive numbers can be raised to the power of 0.
Solution
By convention, 0^0 is often defined as 1 in combinatorics, although it can be considered indeterminate in other contexts.
Correct Answer:
A
— 0^0 is also equal to 1.
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Q. If a^0 = 1 for any non-zero number a, which of the following statements is true?
-
A.
0 raised to any power is also 1.
-
B.
Any number raised to the power of zero is zero.
-
C.
Only positive numbers can be raised to the power of zero.
-
D.
The exponent zero indicates the multiplicative identity.
Solution
The exponent zero indicates the multiplicative identity, meaning any non-zero number raised to the power of zero equals one.
Correct Answer:
D
— The exponent zero indicates the multiplicative identity.
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Q. If a^3 * a^(-2) = a^x, what is the value of x? (2023)
Solution
Using the property of exponents, a^3 * a^(-2) = a^(3 - 2) = a^1, hence x = 1.
Correct Answer:
A
— 1
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Q. If a^3 * b^2 = 64 and a = 2, what is the value of b? (2023)
Solution
Substituting a = 2, we have 2^3 * b^2 = 64, which simplifies to 8b^2 = 64. Thus, b^2 = 8, leading to b = 4.
Correct Answer:
B
— 8
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Q. If a^3 = b^2, which of the following is true?
-
A.
a = b^(2/3)
-
B.
b = a^(3/2)
-
C.
a^2 = b^(3/2)
-
D.
b^3 = a^2
Solution
From a^3 = b^2, we can express b in terms of a as b = a^(3/2).
Correct Answer:
B
— b = a^(3/2)
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Q. If a^m * a^n = a^p, what is the value of p?
-
A.
m + n
-
B.
m - n
-
C.
m * n
-
D.
m / n
Solution
According to the laws of exponents, when multiplying like bases, we add the exponents: p = m + n.
Correct Answer:
A
— m + n
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Q. If a^m * a^n = a^p, which of the following is true?
-
A.
m + n = p
-
B.
m - n = p
-
C.
m * n = p
-
D.
m / n = p
Solution
The property of exponents states that when multiplying like bases, you add the exponents: m + n = p.
Correct Answer:
A
— m + n = p
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Q. If a^x = b^y and a = b, what can be inferred about x and y?
-
A.
x = y
-
B.
x > y
-
C.
x < y
-
D.
x and y are unrelated
Solution
If a = b, then a^x = b^y implies x must equal y for the equality to hold.
Correct Answer:
A
— x = y
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Q. If x = 2 and y = 3, what is the value of 2^(x+y)?
Solution
Substituting x and y gives us 2^(2+3) = 2^5 = 32.
Correct Answer:
B
— 16
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Q. If x = 2 and y = 3, what is the value of x^y + y^x?
Solution
Calculating, we find 2^3 + 3^2 = 8 + 9 = 17.
Correct Answer:
B
— 17
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Q. If x = 2^3 and y = 2^2, what is the value of x/y? (2023)
Solution
We have x = 8 and y = 4. Thus, x/y = 8/4 = 2.
Correct Answer:
A
— 2
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Q. In a certain context, if the expression 5^(x+1) = 125 is true, what is the value of x?
Solution
Since 125 can be expressed as 5^3, we have 5^(x+1) = 5^3, thus x + 1 = 3, leading to x = 2.
Correct Answer:
B
— 2
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Q. In the context of mathematical exponents, which of the following statements is true?
-
A.
a^m * a^n = a^(m+n)
-
B.
a^(m+n) = a^m + a^n
-
C.
a^0 = 1 for any a ≠ 0
-
D.
a^(-n) = 1/a^n
Solution
The correct statements regarding exponents include that a^m * a^n = a^(m+n) and a^(-n) = 1/a^n. However, a^(m+n) = a^m + a^n is incorrect.
Correct Answer:
B
— a^(m+n) = a^m + a^n
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Q. In the context of mathematical expressions, which of the following statements about exponents is true?
-
A.
Exponents can only be positive integers.
-
B.
The product of two numbers with the same base is the sum of their exponents.
-
C.
Exponents can be ignored in calculations.
-
D.
Exponents are irrelevant in algebra.
Solution
The product of two numbers with the same base is indeed the sum of their exponents, as per the laws of exponents.
Correct Answer:
B
— The product of two numbers with the same base is the sum of their exponents.
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Q. In the context of mathematical expressions, which of the following statements best describes the role of exponents?
-
A.
They indicate the number of times a base is multiplied by itself.
-
B.
They are used to denote the addition of two numbers.
-
C.
They represent the square root of a number.
-
D.
They are irrelevant in algebraic equations.
Solution
Exponents indicate how many times a base is multiplied by itself, which is fundamental in understanding powers in mathematics.
Correct Answer:
A
— They indicate the number of times a base is multiplied by itself.
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Q. What is the result of (2^3)^2?
-
A.
2^5
-
B.
2^6
-
C.
2^7
-
D.
2^8
Solution
Using the power of a power property, (a^m)^n = a^(m*n), we have (2^3)^2 = 2^(3*2) = 2^6.
Correct Answer:
B
— 2^6
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Q. What is the result of 5^2 * 5^(-3)? (2023)
-
A.
5^1
-
B.
5^(-1)
-
C.
5^0
-
D.
5^(-5)
Solution
Using the property of exponents, we combine the exponents: 5^(2 + (-3)) = 5^(-1).
Correct Answer:
B
— 5^(-1)
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Q. What is the result of simplifying the expression 2^(3x) * 2^(2x) / 2^(4x)?
-
A.
2^(x)
-
B.
2^(x-1)
-
C.
2^(0)
-
D.
2^(5x)
Solution
Using the properties of exponents, we combine the exponents: (3x + 2x - 4x) = x, thus the result is 2^x.
Correct Answer:
A
— 2^(x)
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Q. What is the result of simplifying the expression 2^(3x) * 2^(2x) / 2^(5x)?
-
A.
2^0
-
B.
2^x
-
C.
2^(3x + 2x - 5x)
-
D.
2^(5x)
Solution
Using the properties of exponents, we combine the exponents: 2^(3x + 2x - 5x) = 2^0 = 1.
Correct Answer:
C
— 2^(3x + 2x - 5x)
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