Fractional Exponents
Exponent Laws
Radicals
Factoring and equations
Distribution/factoring
100
When 
is simplified, the result is 
, where 
and 
are positive integers and is as small as possible. What is 
?
Since the prime factorization of 100 is 
, we cannot simplify 
any further. Therefore, we have 
.
100
What is 
We simplify the expression inside the parentheses first. Simplifying the exponent first, and then subtracting, we get 
Then, the whole expression is 
. Once again simplifying exponents first, we get 
, which gives us 
100
If 
, solve for 
Express your answer in simplest fractional form.
We can start by cross-multiplying:
Checking, we see that this value of does indeed work, so it is not an extraneous solution.
100
Solve for , where 
and 
. Express your answer as a simplified common fraction
can be written as 
. Because must be positive, the only factor that can be considered is 
. Therefore:
100
he ratio 
is closest to which whole number?
200
What is the value of 
?
Simplify under the radical first: 
,
and the cube root of 
is 
.
200
What is the value of the expression 
For any real number , 
It follows that 
Therefore, 
200
If 
, find the average of all the possible values of X
First, we start by squaring both sides of the equation
From here, we can see that the only possible values of are 3 and -3. Therefore the average is 
200
Solve for the largest value of such that 
Express your answer as a simplified common fraction.
Expanding, we have 
Hence, we see that 
Therefore, 
or 
Of these, the greater value for is 
200
Factor the following expression: 
he greatest common factor of 
and 
is . We factor out of both terms to get
300
Simplify: 
300
What is the result when we compute
and
and then add the two results?
Recall that 
. Thus, our second sum can be rewritten as
When we add this with
we can pair the terms conveniently:
Because any number plus its negation is zero, each of these pairs of terms sum to zero, and the sum of the entire sequence is
300
Find the value of that satisfies 
Express your answer in simplest fractional form.
We begin by multiplying out the denominator and then squaring both sides
Checking, we see that this value of satisfies the original equation, so it is not an extraneous solution
300
What is the positive difference of the solutions of 
Factoring the quadratic in the numerator does not look pleasant, so we go ahead and multiply through by the denominator to get
Therefore the solutions are 
and 
which have a difference of 
300
Expand the following expression: 
We apply the distributive property to get
400
What is the difference between the positive square root of 64 and the cube root of 64?
The positive square root of 64 is 
The cube root of 64 is 
The difference is 
400
What is the sum of all positive integer cubes that are less than 
We sum 
. Since 
, there are no more positive integer cubes less than 100
400
solve for a
We can factor a constant out of the first radical:
Then, we can combine like terms and solve:
400
The sum of two numbers and 
is 153, and the value of the fraction 
is 0.7. What is the value of 
We have the system of equations:
From the second equation, multiplying both sides by gives 
. Next, substituting the second equation into the first to eliminate gives 
, or 
. Plugging in this value into the first equation in the original system of equations gives 
or 
. Thus, 
.
400
Compute 
Rearranging the terms, we find that this is equal to 
500
What is the value of 
such that 
We perform the following sequence of substitutions and operations: 
Hence 
, or 
500
Calculate 
Recall that 0 to any positive power is 0. Also recall that 
if is even. Because 5 is positive and 4 is even, we can apply these rules to the given expression to get
500
The square root of is greater than 2 and less than 4. How many integer values of satisfy this condition?
We have: 
. Squaring, we get 
. Thus, the integers from 15 to 5, inclusive, satisfy this inequality. That's a total of 
integers.
500
Let , and 
be positive numbers satisfying 
; 
; and 
. What is the value of ?
Multiplying all three equations together, we find 
, so 
Thus,
500
Completely factor the following expression:
First, we combine like terms in the expression:
We can factor out a 
from the expression, to get