How to find patterns in exponents - ACT Math
Card 0 of 9
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.
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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_
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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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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.
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Evaluate 
Evaluate
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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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