Because we are only multiplying terms in the numerator, we can disregard the parentheses.

To combine like terms in the numerator, we add their exponents.

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

Remember that any negative exponents stay in the denominator.

Simplify the following expression by combining like terms:

Begin by looking for terms to combine. In this case, we only have 2 terms we can combine. Remember, we can only combine terms that have the same exponent and variable.

In this case, the green colors are the only ones which can be combined:

So our answer is:

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms to solve.

and have no like terms and cannot be combined with anything.

5 and -5 can be combined however:

This leaves us with .

When multiplying an odd number of negatives, the answer is negative.

When multiplying an even number of negatives, the answer is positive.

A quadratic trinomial is a perfect square if and only if takes the form

for some values of and .

, so

and .

For to be a perfect square, it must hold that

,

so . This is the correct choice.

First, distribute –5 through the parentheses by multiplying both terms by –5.

Then, combine the like-termed variables (–5x and –3x).

Simplify the numerator by multiplying in the term

Cancel out like terms in the numerator and denominator.

When simplifying an equation,you must find a common factor for all values in the equation, including both sides.

and, can all be divided by so divide them all at once

.

This leaves you with

.

To expand, multiply 8x by both terms in the expression (3x + 7).

8x multiplied by 3x is 24x².

8x multiplied by 7 is 56x.

Therefore, 8x(3x + 7) = 24x² + 56x.

The domain of a function consists of all of the values of x for which f(x) is defined. When determining the domain of a funciton, the three most important things we want to consider are square roots, logarithms, and denominators of fractions. These tend to signal places where the function is not defined.

First, let's look at the term. Remember we can only find the square root of nonnegative values. Thus, everything under the square root symbol must be greater than or equal to zero. This tells us that, for this function, .

Second, we need to look at the natural logarithm. The natural logarithm can only be applied to positive numbers (which don't include zero). Thus, everything within the paranethesis of the natural logarithm must be greater than zero.

There are several ways to solve this inequality. One way is to factor the left side and examine the factors. We know that because of the difference of squares factorization formula.

.

This statement will only be true in two situations; either both factors must be positive, or both must be negative.

We can see that if , then the factor will be positive, but the factor will be negative. If we were to multiply a negative and a positive number, we would get a negative number. Thus, is not larger than zero when .

Let's consider the interval . In this case, both and would be positive. Thus, when .

Third, consider the interval . In this case, the first factor will be negative, and the second will be positive, so their product would be negative, and would not be greater than zero.

To summarize, only if .

We can see now that f(x) is only defined if and .

There is one more piece of information we need to consider--the denominator of f(x). Remember that a fraction is not defined if its denominator equals zero. Thus, if the denominator is equal to zero at a certain value of x, we can't include this value of x in the domain of f(x).

We can set the denominator equal to zero and solve to see if there are any values of x where the denominator would be zero.

Rewrite this as an exponential equation. In general, the equation can be rewritten as , provided that a is positive.

If we put into exponential form, we obtain

We can solve this for x.

So, let's put all of this information together. We know that f(x) is only defined if ALL of these conditions are met:

The only interval for which this is true is if x is greater than (and not equal to) zero but less than (and not equal to) 1. Thus, the domain of f(x) is .

The answer is .