Exponents - GMAT Quantitative
Card 1 of 104
is an integer. Is there a real number 
such that 
?
Statement 1: 
is negative
Statement 2: 
is even
Statement 1:
Statement 2:
Tap to reveal answer
The equivalent question is "does 
have a real 
root?"
If you know only that 
is negative, you need to know whether 
is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.
If you know only that 
is even, you need to know whether 
is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
The equivalent question is "does
If you know only that
If you know only that
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
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Which is the greater quantity, 
or 
- or are they equal?
Statement 1: 
Statement 2: 
Which is the greater quantity,
Statement 1:
Statement 2:
Tap to reveal answer
From Statement 1 alone,
Now assume Statement 2 alone. We show that this is insufficient with two cases:
Case 1: 
; 
; therefore, 
Case 1: 
; 
; therefore, 
From Statement 1 alone,
Now assume Statement 2 alone. We show that this is insufficient with two cases:
Case 1:
Case 1:
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Does 
exist?
Statement 1: 
and 
are both negative.
Statement 2: 
divided by 2 yields an integer.
Does
Statement 1:
Statement 2:
Tap to reveal answer
A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then 
, being the product of two negative numbers, must be positive, and 
exists.
Statement 2 is irrelevant; 4 and 
both yield integers when divided by 2, but 
and 
does not exist.
A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then
Statement 2 is irrelevant; 4 and
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Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.
Johnny filled in both shapes with numbers. Did he succeed?
Statement 1: He filled in the circle with the number "10".
Statement 2: He filled in the square with a negative integer.
Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.
Johnny filled in both shapes with numbers. Did he succeed?
Statement 1: He filled in the circle with the number "10".
Statement 2: He filled in the square with a negative integer.
Tap to reveal answer
The number 
is a number written in scientific notation if and only of two conditions are true:
-
-
is an integer
By Statement 1 Johnny filled in the circle incorrectly, since it makes 
.
By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.
The number
-
-
By Statement 1 Johnny filled in the circle incorrectly, since it makes
By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.
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Solve the following rational expression:
(1) 
(2) 
Solve the following rational expression:
(1)
(2)
Tap to reveal answer
When replacing m=5 in the expression we get:
Therefore statement (1) ALONE is not sufficient.
When replacing m=2n in the expression we get:
Therefore statement (2) ALONE is sufficient.
When replacing m=5 in the expression we get:
Therefore statement (1) ALONE is not sufficient.
When replacing m=2n in the expression we get:
Therefore statement (2) ALONE is sufficient.
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Is 
?
(1) 
(2) 
Is
(1)
(2)
Tap to reveal answer
From statement 1 we get that 
and 
.
So the first term is positive and the second term is negative, which means that 
is negative; therefore the statement 1 alone allows us to answer the question.
Statement 2 tells us that 
. If 
, we have 
which is less than 
. Therefore in this case 
.
For 
, we have 
which is greater than 
. So in this case 
.
So statement 2 is insufficient.
Therefore the correct answer is A.
From statement 1 we get that
So the first term is positive and the second term is negative, which means that
Statement 2 tells us that
For
So statement 2 is insufficient.
Therefore the correct answer is A.
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Note: Figure NOT drawn to scale.
Examine the above diagram. True or false: 
.
Statement 1: 
Statement 2: 
and 
have the same perimeter.
Note: Figure NOT drawn to scale.
Examine the above diagram. True or false:
Statement 1:
Statement 2:
Tap to reveal answer
From Statement 1 alone, it follows by the similarity of the triangles that 
. These are congruent inscribed angles of a circle, which intercept congruent arcs, so 
. Since congruent arcs have congruent chords, 
.
Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.
From Statement 1 alone, it follows by the similarity of the triangles that
Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.
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is a nonzero number. Is it negative or positive?
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
Tap to reveal answer
Both statements together are inufficient to produce an answer. For example,
If 
, then 
and 
.
If 
, then 
and 
.
In both cases, and , but the signs of 
differ between cases.
Both statements together are inufficient to produce an answer. For example,
If
If
In both cases,
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is a nonzero number. Is it negative or positive?
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
Tap to reveal answer
All negative numbers are less than their (positive) squares, as are all positive numbers greater than 1. Therefore, if Statement 1 is assumed, 
.
can be determined to be positive.
Statement 2 alone is inconclusive. For example, if 
, then 
, and if 
, 
. In both cases, 
, but 
has different signs in the two cases.
All negative numbers are less than their (positive) squares, as are all positive numbers greater than 1. Therefore, if Statement 1 is assumed,
Statement 2 alone is inconclusive. For example, if
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Philip has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.
.
Did Philip succeed?
Statement 1: Philip wrote 
in the circle.
Statement 2: Philip wrote 
in the square.
Philip has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.
Did Philip succeed?
Statement 1: Philip wrote
Statement 2: Philip wrote
Tap to reveal answer
A number in scientific notation takes the form
where 
and 
is an integer of any sign.
Statement 1 alone proves that Philip entered a correct number into the circle, since 
. Statement 2 alone proves that he entered a correct number into the square, since 
is an integer. But each statement alone is insufficient, since each leaves uinclear whether the other number is valid. The two statements together, however, prove that Philip put correct numbers in both places, thereby writing a number in scientific notation.
A number in scientific notation takes the form
where
Statement 1 alone proves that Philip entered a correct number into the circle, since
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Myoshi has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.
.
Did Myoshi succeed?
Statement 1: Myoshi wrote 
in the circle.
Statement 2: Myoshi wrote 
in the square.
Myoshi has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.
Did Myoshi succeed?
Statement 1: Myoshi wrote
Statement 2: Myoshi wrote
Tap to reveal answer
A number in scientific notation takes the form
where 
and 
is an integer of any sign.
Assuming Statement 1 alone, Myoshi did not succeed, since she entered an incorrect number into the circle - 
.
Statement 2 alone is inconclusive. Myoshi entered a correct number into the square, since 
is an integer. But the question is open, since it is not known whether she entered a correct number into the circle or not.
A number in scientific notation takes the form
where
Assuming Statement 1 alone, Myoshi did not succeed, since she entered an incorrect number into the circle -
Statement 2 alone is inconclusive. Myoshi entered a correct number into the square, since
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is a number not in the set 
.
Of the elements 
, which is the greatest?
Statement 1: 
is a negative number.
Statement 2: 
Of the elements
Statement 1:
Statement 2:
Tap to reveal answer
Statement 1 alone is inconclusive, as can be demonstrated by examining two negative values of 
other than 
.
Case 1: 
.
Then
is the greatest of these values.
Case 2: 
Then
is the greatest of these values.
Now assume Statement 2 alone. Either 
or 
.
Case 1: 
.
Then 
, so 
; similarly, 
.
is the greatest of the three.
Case 2: 
.
Odd power 
is negative, and even powers and 
are positive, so one of the latter two is the greatest. Since 
, it follows that 
. It then follows that 
, or 
.
Again, is the greatest of the three.
Statement 2 alone is sufficient, but not Statement 1.
Statement 1 alone is inconclusive, as can be demonstrated by examining two negative values of
Case 1:
Then
Case 2:
Then
Now assume Statement 2 alone. Either
Case 1:
Then
Case 2:
Odd power
Again,
Statement 2 alone is sufficient, but not Statement 1.
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is a number not in the set 
.
Of the elements 
, which is the greatest?
Statement 1: 
is a negative number.
Statement 2: 
Of the elements
Statement 1:
Statement 2:
Tap to reveal answer
Assume both statements are known. The greatest of the three numbers must be 
or 
, since even powers of negative numbers are positive and odd powers of negative numbers are negative.
Case 1: 
Case 2: 
,
then
In both cases, 
is negative and , but in one case, 
is the greatest number, and in the other, 
is. The two statements together are inconclusive.
Assume both statements are known. The greatest of the three numbers must be
Case 1:
Case 2:
then
In both cases,
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is an integer. Is there a real number such that ?
Statement 1: is negative
Statement 2: is even
Statement 1:
Statement 2:
Tap to reveal answer
The equivalent question is "does have a real root?"
If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.
If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
The equivalent question is "does
If you know only that
If you know only that
If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.
Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.
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Which is the greater quantity, or - or are they equal?
Statement 1:
Statement 2:
Which is the greater quantity,
Statement 1:
Statement 2:
Tap to reveal answer
From Statement 1 alone,
Now assume Statement 2 alone. We show that this is insufficient with two cases:
Case 1:
; ; therefore,
Case 1:
; ; therefore,
From Statement 1 alone,
Now assume Statement 2 alone. We show that this is insufficient with two cases:
Case 1:
Case 1:
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Does exist?
Statement 1: and are both negative.
Statement 2: divided by 2 yields an integer.
Does
Statement 1:
Statement 2:
Tap to reveal answer
A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then , being the product of two negative numbers, must be positive, and exists.
Statement 2 is irrelevant; 4 and both yield integers when divided by 2, but and does not exist.
A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then
Statement 2 is irrelevant; 4 and
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Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.
Johnny filled in both shapes with numbers. Did he succeed?
Statement 1: He filled in the circle with the number "10".
Statement 2: He filled in the square with a negative integer.
Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.
Johnny filled in both shapes with numbers. Did he succeed?
Statement 1: He filled in the circle with the number "10".
Statement 2: He filled in the square with a negative integer.
Tap to reveal answer
The number is a number written in scientific notation if and only of two conditions are true:
-
-
is an integer
By Statement 1 Johnny filled in the circle incorrectly, since it makes .
By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.
The number
-
-
By Statement 1 Johnny filled in the circle incorrectly, since it makes
By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.
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Solve the following rational expression:
(1)
(2)
Solve the following rational expression:
(1)
(2)
Tap to reveal answer
When replacing m=5 in the expression we get:
Therefore statement (1) ALONE is not sufficient.
When replacing m=2n in the expression we get:
Therefore statement (2) ALONE is sufficient.
When replacing m=5 in the expression we get:
Therefore statement (1) ALONE is not sufficient.
When replacing m=2n in the expression we get:
Therefore statement (2) ALONE is sufficient.
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Is ?
(1)
(2)
Is
(1)
(2)
Tap to reveal answer
From statement 1 we get that and .
So the first term is positive and the second term is negative, which means that is negative; therefore the statement 1 alone allows us to answer the question.
Statement 2 tells us that . If , we have which is less than . Therefore in this case .
For , we have which is greater than . So in this case .
So statement 2 is insufficient.
Therefore the correct answer is A.
From statement 1 we get that
So the first term is positive and the second term is negative, which means that
Statement 2 tells us that
For
So statement 2 is insufficient.
Therefore the correct answer is A.
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Note: Figure NOT drawn to scale.
Examine the above diagram. True or false: .
Statement 1:
Statement 2: and have the same perimeter.
Note: Figure NOT drawn to scale.
Examine the above diagram. True or false:
Statement 1:
Statement 2:
Tap to reveal answer
From Statement 1 alone, it follows by the similarity of the triangles that . These are congruent inscribed angles of a circle, which intercept congruent arcs, so . Since congruent arcs have congruent chords, .
Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.
From Statement 1 alone, it follows by the similarity of the triangles that
Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.
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