Pattern Behaviors in Exponents - GRE Quantitative Reasoning
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Jack has 
,
to invest. If he invests two-thirds of it into a high-yield savings account with an annual interest rate of 
, compounded quarterly, and the other third in a regular savings account at 
simple interest, how much does Jack earn after one year?
Jack has
First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).
Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:
Plug in the values given:
Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:
Add the two together, and we see that Jack makes a total of, 
off of his investments.
First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).
Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:
Plug in the values given:
Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:
Add the two together, and we see that Jack makes a total of,
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A five-year bond is opened with 
in it and an interest rate of 
%, compounded annually. This account is allowed to compound for five years. Which of the following most closely approximates the total amount in the account after that period of time?
A five-year bond is opened with
Each year, you can calculate your interest by multiplying the principle (
) by 
. For one year, this would be:
For two years, it would be:
, which is the same as 
Therefore, you can solve for a five year period by doing:
Using your calculator, you can expand the 
into a series of multiplications. This gives you 
, which is closest to 
.
Each year, you can calculate your interest by multiplying the principle (
For two years, it would be:
Therefore, you can solve for a five year period by doing:
Using your calculator, you can expand the
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If a cash deposit account is opened with 
for a three year period at 
% interest compounded once annually, which of the following is closest to the positive difference between the interest accrued in the third year and the interest accrued in the second year?
If a cash deposit account is opened with
It is easiest to break this down into steps. For each year, you will multiply by 
to calculate the new value. Therefore, let's make a chart:
After year 1: 
; Total interest: 
After year 2: 
; Let us round this to 
; Total interest: 
After year 3: 
; Let us round this to 
; Total interest: 
Thus, the positive difference of the interest from the last period and the interest from the first period is: 
It is easiest to break this down into steps. For each year, you will multiply by
After year 1:
After year 2:
After year 3:
Thus, the positive difference of the interest from the last period and the interest from the first period is:
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What digit appears in the units place when 
is multiplied out?
What digit appears in the units place when
This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.
Observe the first few powers of 2:
21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64, 27 = 128, 28 = 256 . . .
The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.
The second number in the sequence is 4, so the answer is 4.
This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.
Observe the first few powers of 2:
21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64, 27 = 128, 28 = 256 . . .
The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.
The second number in the sequence is 4, so the answer is 4.
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If 
, then which of the following must also be true?
If
We know that the expression must be negative. Therefore one or all of the terms x7, y8 and z10 must be negative; however, even powers always produce positive numbers, so y8 and z10 will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x7 must be negative, so x must be negative. Thus, the answer is x < 0.
We know that the expression must be negative. Therefore one or all of the terms x7, y8 and z10 must be negative; however, even powers always produce positive numbers, so y8 and z10 will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x7 must be negative, so x must be negative. Thus, the answer is x < 0.
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Quantitative Comparison
Quantity A: _x_2
Quantity B: _x_3
Quantitative Comparison
Quantity A: _x_2
Quantity B: _x_3
Let's pick numbers. For quantitative comparisons with exponents, it's good to try 0, a negative number, and a fraction.
0: 02 = 0, 03 = 0, so the two quantities are equal.
–1: (–1)2 = 1, (–1)3 = –1, so Quantity A is greater.
Already we have a contradiction so the answer cannot be determined.
Let's pick numbers. For quantitative comparisons with exponents, it's good to try 0, a negative number, and a fraction.
0: 02 = 0, 03 = 0, so the two quantities are equal.
–1: (–1)2 = 1, (–1)3 = –1, so Quantity A is greater.
Already we have a contradiction so the answer cannot be determined.
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Which quantity is the greatest?
Quantity A
Quantity B
Which quantity is the greatest?
Quantity A
Quantity B
First rewrite quantity B so that it has the same base as quantity A.
can be rewriten as 
, which is equivalent to 
.
Now we can compare the two quantities.
is greater than 
.
First rewrite quantity B so that it has the same base as quantity A.
Now we can compare the two quantities.
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Which of the following is a multiple of 
?
Which of the following is a multiple of
For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:
Now, in order for you to have a number that is a multiple of this, you will need to have at least 
in the prime factorization of the given number. For each of the answer choices, you have:
; This is the answer.
For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:
Now, in order for you to have a number that is a multiple of this, you will need to have at least
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Simplify the following:
Simplify the following:
Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the 
into its prime factors:
Note that these have a common factor of 
. Therefore, you can rewrite the numerator as:
Now, put this back into your fraction:
Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the
Note that these have a common factor of
Now, put this back into your fraction:
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Simplify the following:
Simplify the following:
With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:
Numerator
Continuing the simplification:
Now, these factors have in common a 
. Factor this out:
Denominator
This is much simpler:
Now, return to your fraction:
Cancel out the common factors of 
:
With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:
Numerator
Continuing the simplification:
Now, these factors have in common a
Denominator
This is much simpler:
Now, return to your fraction:
Cancel out the common factors of
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