# GMAT Math : Understanding exponents

## Example Questions

### Example Question #31 : Understanding Exponents

. Order from least to greatest: .

This is impossible, since at least one expression is undefined.

Explanation:

It might be easiest to order  first and work from there.

Since  is a positive number less than 1, if , then . Therefore, those four expressions, from least to greatest, are

.

If , then  - that is, . So changing the signs of the exponents reverses the order. As a result, the orginal four expressions, in ascending order, are

### Example Question #32 : Understanding Exponents

. Order from least to greatest: .

Explanation:

If  is negative, then:

Even powers  and  are positive, with  and

Since ,

It follows that  , and .

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

.

Odd powers  and  are negative, with  and .

, so , and

As before, changing their exponents to their opposites reverses the order:

Setting the negative numbers less than the positive numbers:

.

### Example Question #31 : Understanding Exponents

. Order from least to greatest: .

Explanation:

Since  is a positive number less than 1, if , then . Therefore,

.

### Example Question #32 : Understanding Exponents

Simplify.

Explanation:

In order to solve this problem we have to keep in mind three properties of exponents:

The first thing we can do is take care of the negative exponent:

I included a  as the exponent for  in order to make the calculation easier to see.

We can now simplify by looking into each variable individually:

The last step is to put these answers together:

### Example Question #33 : Exponents

Simplify.

Explanation:

In order to solve this problem we have to keep in mind three properties of exponents:

The first thing we can do is take care of the negative exponents:

We can now simplify by working on each variable individually:

Don't forget to simplify the coefficients as well:

We can now arrive to our answer by combining these:

### Example Question #36 : Understanding Exponents

Which of the following numbers is in scientific notation?

Explanation:

The number in front must have an absolute value greater than or equal to 1, and less than 10. Of these choices, only  qualifies.

### Example Question #1141 : Problem Solving Questions

. Order from least to greatest: .

Explanation:

If , then if . Therefore,

.

### 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!