Exponential Operations - GRE Quantitative Reasoning
Card 1 of 256
, and 
is odd.
Quantity A: 
Quantity B: 
Quantity A:
Quantity B:
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The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
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Simplify: 32 * (423 - 421)
Simplify: 32 * (423 - 421)
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Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
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Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
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We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
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, and is odd.
Quantity A:
Quantity B:
Quantity A:
Quantity B:
Tap to reveal answer
The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
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Simplify: 32 * (423 - 421)
Simplify: 32 * (423 - 421)
Tap to reveal answer
Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
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Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
Tap to reveal answer
We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
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, and is odd.
Quantity A:
Quantity B:
Quantity A:
Quantity B:
Tap to reveal answer
The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
The first thing to note is the relationship between (–b) and (1 – b):
(–b) < (1 – b) because (–b) + 1 = (1 – b).
Now when b > 1, (1 – b) < 0 and –b < 0. Therefore (–b) < (1 – b) < 0.
Raising a negative number to an odd power produces another negative number.
Thus (–b)a < (1 – b)a < 0.
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Simplify: 32 * (423 - 421)
Simplify: 32 * (423 - 421)
Tap to reveal answer
Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
Begin by noting that the group (423 - 421) has a common factor, namely 421. You can treat this like any other constant or variable and factor it out. That would give you: 421(42 - 1). Therefore, we know that:
32 * (423 - 421) = 32 * 421(42 - 1)
Now, 42 - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:
32 * 421(42 - 1) = 32 * 421(3 * 5)
Combining multiples withe same base, you get:
33 * 421 * 5
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Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
Quantitative Comparison
Quantity A: 64 – 32
Quantity B: 52 – 42
Tap to reveal answer
We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
We can solve this without actually doing the math. Let's look at 64 vs 52. 64 is clearly bigger. Now let's look at 32 vs 42. 32 is clearly smaller. Then, bigger – smaller is greater than smaller – bigger, so Quantity A is bigger.
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Simplify
Simplify
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Divide the coefficients and subtract the exponents.
Divide the coefficients and subtract the exponents.
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Which of the following is equal to the expression 
, where
xyz ≠ 0?
Which of the following is equal to the expression , where
xyz ≠ 0?
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(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.
(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.
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Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.
Quantity A Quantity B
(23 )2 (22 )3
Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.
Quantity A Quantity B
(23 )2 (22 )3
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The two quantites are equal. To take the exponent of an exponent, the two exponents should be multiplied.
(23 )2 or 23*2 = 64
(22 )3 or 22*3 = 64
Both quantities equal 64, so the two quantities are equal.
The two quantites are equal. To take the exponent of an exponent, the two exponents should be multiplied.
(23 )2 or 23*2 = 64
(22 )3 or 22*3 = 64
Both quantities equal 64, so the two quantities are equal.
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Simplify: y3x4(yx3 + y2x2 + y15 + x22)
Simplify: y3x4(yx3 + y2x2 + y15 + x22)
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When you multiply exponents, you add the common bases:
y4 x7 + y5x6 + y18x4 + y3x26
When you multiply exponents, you add the common bases:
y4 x7 + y5x6 + y18x4 + y3x26
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Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.
Quantity A: 
Quantity B: 
Indicate whether Quantity A or Quantity B is greater, or if they are equal, or if there is not enough information given to determine the relationship.
Quantity A:
Quantity B:
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By using exponent rules, we can simplify Quantity B.
Also, we can simplify Quantity A.
Since n is positive, 
By using exponent rules, we can simplify Quantity B.
Also, we can simplify Quantity A.
Since n is positive,
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If 
, what is the value of 
?
If
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Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.
Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.
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If 
, what is the value of 
?
If
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Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).
The term on the right can be rewritten, as 27 is equal to 3 to the third power.
Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.
We now know that the exponents must be equal, and can solve for 
.
Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).
The term on the right can be rewritten, as 27 is equal to 3 to the third power.
Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.
We now know that the exponents must be equal, and can solve for
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Simplify 
.
Simplify
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First, simplify 
by adding the exponents to get 
.
Then simplify 
by multiplying the exponents to get 
.
This gives us 
. We cannot simplify any further.
First, simplify
Then simplify
This gives us
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Simplify 
.
Simplify
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Start by simplifying each individual term between the plus signs. We can add the exponents in 
and 
so each of those terms becomes 
. Then multiply the exponents in 
so that term also becomes 
. Thus, we have simplified the expression to 
which is 
.
Start by simplifying each individual term between the plus signs. We can add the exponents in
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If 
, what is the value of 
If
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To attempt this problem, note that 
.
Now note that when multiplying numbers, if the base is the same, we may add the exponents:
This can in turn be written in terms of nine as follows (recall above)
To attempt this problem, note that
Now note that when multiplying numbers, if the base is the same, we may add the exponents:
This can in turn be written in terms of nine as follows (recall above)
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If 
, what is the value of 
If
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When dealing with exponenents, when multiplying two like bases together, add their exponents:
However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:
When dealing with exponenents, when multiplying two like bases together, add their exponents:
However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:
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