GMAT Math : Algebra

Example Questions

Example Question #1142 : Problem Solving Questions

. Order from least to greatest: .

Explanation:

If  is negative, then even powers  and  are positive, with  and

Since ,

It follows that  , and .

Odd powers  and  are negative, with  and .

, so , and .

Therefore,

.

Example Question #61 : Algebra

. Order from least to greatest: .

Explanation:

Since  is a positive number less than , if , then . Therefore,

If , then —that is, . So, changing the signs of the exponents reverses the order. As a result,

Example Question #36 : Exponents

Simplify the following expression:

Explanation:

This problem may look difficult, but it can be easily solved through application of three basic exponent rules.

1)

This states that when we are dividing exponents of the same base, we can combine them by subtracting the exponents.

2)

This is just the opposite idea, saying that when we multiply two or more exponents of the same base, we can combine them by adding the exponents.

3)

This one is often confused with the second rule, but it is important to keep them straight. When raising an exponent to another power, simply multiply the exponents together.

With those in mine, let's return to our initial equation. The first step is highlighted in red.

Begin by distributing that exponent, because then, it allows us to start simplifying everything else. By doing that we get the following:

All we did there was multiply each of those exponents by

Next, we will use Rule 1 from above, to start simplifying each fraction. For starters, let's take the 's in the first fraction.

Because we are dividing, we will subtract our exponents to get positive . Because it is positive, we will keep in in the numerator and cancel out the 's in the denominator.

Next, we do the same with the y's and z's in the first fraction to get the following.

Now, we multiply the two fractions together to get our answer. We will add the exponents of the two 's together, as well as the two 's. We will save the 's for the final step.

Finally, we do a final subtraction step with our 's. Our exponent for  on the top is really a , and , so our  exponent is . If you are worried because none of our answer choices have a negative exponent, don't be. Simply put the  in the denominator and you are good to go!

Example Question #1141 : Problem Solving Questions

Explanation:

If we add the same number with the same exponent four times, we can see the result will be four times the value. Assuming we couldn't recognize this at first glance, we can factor the common term our of our expression:

We cannot just simply multiply our bases together, however, because one of them has an exponent greater than 1. What we must do, then, is express 4 in a way that makes our two bases equal, which will allow us to add their exponents to arrive at our final answer:

Example Question #1141 : Problem Solving Questions

Explanation:

As with any binomial with an exponent of 2, we can FOIL this expression to find its most simple form. As the expression is squared, we could rewrite it in the following way:

Now to FOIL the expression, we multiply the First terms, add the product of the Outside terms, then the product of the Inside terms, and finally the product of the Last terms, hence the acronym FOIL:

Example Question #61 : Algebra

Simplify the following expression:

Explanation:

Here we need to recall a few simple properties of exponents.

I)When we are dividing exponents of like base, we subtract the exponents.

II)When we are multiplying exponents of like base, we add the exponents.

III)When we are raising exponents to higher powers, we multiply the exponents.

I will apply and explain each of these properties as I solve this problem.

We begin with this:

I will begin by applying property II to the numerator of our fraction. Our like base is x, and the exponents we are adding are 3 and 4, so we get this:

Next we will apply property I to our fraction. Our like base is still x, but our exponents we are subtracting are 7 and 2, so we get:

Finally we apply property III to get our answer:

Example Question #1143 : Problem Solving Questions

Simplify the following expression:

Explanation:

In order to simplify , first simplify each term:

Therefore,

Example Question #41 : Understanding Exponents

Fill in the circle with a number so that this polynomial is prime:

None of the other choices gives a correct answer.

Explanation:

If  is not prime, it is factorable as

where  and .

Therefore, we are looking for a whole number that is not the sum of two factors of 60. The integers that are such a sum are

Of the choices, only 18 is not a sum of factors of 60. It is the correct choice.

Example Question #1152 : Problem Solving Questions

Simplify the following expression:

Explanation:

To simplify the expression start by simplifying each term.

From here, combine like terms.

Example Question #41 : Exponents

Simplify the following expression: