# GMAT Math : Algebra

## Example Questions

### Example Question #21 : Understanding Exponents

Which of the following is equal to   ?

Possible Answers:

Correct answer:

Explanation:

We'll need to remember a few logarithmic properties to answer this question:

Now we can use these same rules to rewrite the log in question:

### Example Question #22 : Understanding Exponents

Simplify the following expression:

.

Possible Answers:

Correct answer:

Explanation:

We start by simplifying the expression on the top. Let's add the exponents inside the parentheses and then multiply by the exponent outside of the parentheses.

Then we substract the denominator exponent form the numerator exponent:

### Example Question #21 : Exponents

Simplify the following:

.

Possible Answers:

Correct answer:

Explanation:

First, we need to add the exponents of the elements with the same base that are multiplied, and subtract the exponents of same-base elements that are divided:

Then

Any value raised to the power of 0 equals 1 so the final result is 1.

### Example Question #31 : Understanding Exponents

. Order from least to greatest: .

Possible Answers:

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

Correct answer:

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: .

Possible Answers:

Correct answer:

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 #1136 : Problem Solving Questions

. Order from least to greatest: .

Possible Answers:

Correct answer:

Explanation:

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

.

### Example Question #31 : Exponents

Simplify.

Possible Answers:

Correct answer:

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 #31 : Exponents

Simplify.

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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: .

Possible Answers:

Correct answer:

Explanation:

If , then if . Therefore,

.