Pattern Behaviors in Exponents - ACT Math
Card 0 of 19
Michelle makes a bank deposit of $1,500 at 4.2% annual interest, compounded monthly. Approximately how much money will be in Michelle’s bank account in 3 years?
Michelle makes a bank deposit of $1,500 at 4.2% annual interest, compounded monthly. Approximately how much money will be in Michelle’s bank account in 3 years?
The formula to use for compounded interest is
where P is the principal (original) amount, r is the interest rate (expressed in decimal form), n is the number of times per year the interest compounds, and t is the total number of years the money is left in the bank. In this problem, P=1,500, r=0.042, n=12, and t=3.
By plugging in, we find that Michelle will have about $1,701 at the end of three years.
The formula to use for compounded interest is
where P is the principal (original) amount, r is the interest rate (expressed in decimal form), n is the number of times per year the interest compounds, and t is the total number of years the money is left in the bank. In this problem, P=1,500, r=0.042, n=12, and t=3.
By plugging in, we find that Michelle will have about $1,701 at the end of three years.
Compare your answer with the correct one above
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,
Compare your answer with the correct one above
Ashley makes a bank deposit of 
at 
annual interest, compounded monthly. About how many years will it take her deposit to grow to 
?
Ashley makes a bank deposit of
The formula for compound interest is
where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.
In this problem, we are solving for time, t. The given variables from the problem are:
Plugging these into the equation above, we get
This simplifies to
We can solve this by taking the natural log of both sides
The formula for compound interest is
where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.
In this problem, we are solving for time, t. The given variables from the problem are:
Plugging these into the equation above, we get
This simplifies to
We can solve this by taking the natural log of both sides
Compare your answer with the correct one above
Alice wants to invest money such that in 
years she has 
. The interest rate is 
compounded quarterly. How much should she invest?
Alice wants to invest money such that in
The formula for compound interest is
where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.
In this problem, we are solving for the principal, P. The given variables from the problem are:
Plugging these into the equation above, we get
Solving for P, we get
The formula for compound interest is
where P is the principal (original) amount, r is the interest rate (in decimal form), n is the number of times per year the interest compounds, t is the time in years, and A is the final amount.
In this problem, we are solving for the principal, P. The given variables from the problem are:
Plugging these into the equation above, we get
Solving for P, we get
Compare your answer with the correct one above
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
Compare your answer with the correct one above
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:
Compare your answer with the correct one above
If an account has interest compounded annually at a rate of 
, what is the balance of the account after 
years of compounding if the initial balance is 
? Round to the nearest cent.
If an account has interest compounded annually at a rate of
Recall that the equation for compounded interest (with annual compounding) is:
Where 
is the balance, 
is the rate of interest, and 
is the number of years.
Thus, for our data, we need to know:
This is approximately 
.
Recall that the equation for compounded interest (with annual compounding) is:
Where
Thus, for our data, we need to know:
This is approximately
Compare your answer with the correct one above
If an account has interest compounded annually at a rate of 
, what is the balance of the account after 
years of compounding if the initial balance is 
? Round to the nearest cent.
If an account has interest compounded annually at a rate of
Recall that the equation for compounded interest (with annual compounding) is:
Where 
is the balance, 
is the rate of interest, and 
is the number of years.
Thus, for our data, we need to know:
This is approximately 
.
Recall that the equation for compounded interest (with annual compounding) is:
Where
Thus, for our data, we need to know:
This is approximately
Compare your answer with the correct one above
If an account has interest compounded quarterly at an annual rate of 
, what is the balance of the account after 
years of compounding if the initial balance is 
? Round to the nearest cent.
If an account has interest compounded quarterly at an annual rate of
Recall that the equation for compounded interest (with quarterly compounding) is:
Where 
is the balance, 
is the rate of interest, 
is the number of years, and 
is the number of times it is compounded per year.
Thus, for our data, we need to know:
This is approximately 
.
Recall that the equation for compounded interest (with quarterly compounding) is:
Where
Thus, for our data, we need to know:
This is approximately
Compare your answer with the correct one above
An account is compounded at a given rate of interest annually for 
years. What is this rate if the beginning balance for the account was 
and its ending balance 
? Round to the nearest hundredth of a percent.
An account is compounded at a given rate of interest annually for
Recall that the equation for compounded interest (with annual compounding) is:
Where is the balance, is the rate of interest, and is the number of years.
Thus, for our data, we need to know:
Now, let's use 
for 
. This gives us:
Using a logarithm, this can be rewritten:
This can be rewritten:
Now, you can solve for 
:
or
Now, finally you can rewrite this as:
Thus, 
Now, round this to 
and recall that 
Thus, 
and 
or 
Recall that the equation for compounded interest (with annual compounding) is:
Where
Thus, for our data, we need to know:
Now, let's use
Using a logarithm, this can be rewritten:
This can be rewritten:
Now, you can solve for
or
Now, finally you can rewrite this as:
Thus,
Now, round this to
Thus,
Compare your answer with the correct one above
What is the value of the expression 
, given that 
?
What is the value of the expression
First, calculate the value of m by taking the two-thirds root of both m3/2 and 64, leaving m=16. Then solve the expression by plugging in 16 for m. This gives 162 + 3(16) = 256 + 48 = 304.
First, calculate the value of m by taking the two-thirds root of both m3/2 and 64, leaving m=16. Then solve the expression by plugging in 16 for m. This gives 162 + 3(16) = 256 + 48 = 304.
Compare your answer with the correct one above
A professional football player’s contract states that he will earn a salary of $1 million his first year. He would then have a 15% increase every year thereafter for the next 5 years. What would he make in his 6th and final season on the contract?
A professional football player’s contract states that he will earn a salary of $1 million his first year. He would then have a 15% increase every year thereafter for the next 5 years. What would he make in his 6th and final season on the contract?
We can represent this as an exponential equation (just use million as a label and not a variable):
$1_m_ * (1.15)5 = $2.01_m_
We can represent this as an exponential equation (just use million as a label and not a variable):
$1_m_ * (1.15)5 = $2.01_m_
Compare your answer with the correct one above
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.
Compare your answer with the correct one above
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.
Compare your answer with the correct one above
For which of the following values of n will (**–**3)n represent a real number between 0 and 1?
For which of the following values of n will (**–**3)n represent a real number between 0 and 1?
In order to transform a negative integer to a positive one, it must be taken to an even power, and to make it a fraction between 0 and 1, it must be taken to a negative power. The only choice that fits both criteria is -2.
In order to transform a negative integer to a positive one, it must be taken to an even power, and to make it a fraction between 0 and 1, it must be taken to a negative power. The only choice that fits both criteria is -2.
Compare your answer with the correct one above
Evaluate 
Evaluate
Compare your answer with the correct one above
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
Compare your answer with the correct one above
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:
Compare your answer with the correct one above
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
Compare your answer with the correct one above