# GRE Math : Exponents and Rational Numbers

## Example Questions

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### Example Question #22 : Algebra

find x

8x=2x+6

3

2

-1

4

2 or -1

3

Explanation:

8 = 23

(23)x = 23x

23x = 2x+6  <- when the bases are the same, you can set the exponents equal to each other and solve for x

3x=x+6

2x=6

x=3

### Example Question #1 : Exponents And Rational Numbers

Compare  and .

The relationship cannot be determined from the information given.

Explanation:

First rewrite the two expressions so that they have the same base, and then compare their exponents.

Combine exponents by multiplying:

This is the same as the first given expression, so the two expressions are equal.

### Example Question #1 : Exponents And Rational Numbers

Solve for

Explanation:

can be written as

Since there is a common base of , we can say

or

### Example Question #1 : How To Find An Exponent From A Rational Number

Solve for .

Explanation:

The basees don't match.

However:

thus we can rewrite the expression as .

Anything raised to negative power means  over the base raised to the postive exponent.

So,

### Example Question #1 : How To Find An Exponent From A Rational Number

Solve for .

Explanation:

The bases don't match.

However:

and we recognize that .

Anything raised to negative power means  over the base raised to the postive exponent.

.

### Example Question #1 : How To Find An Exponent From A Rational Number

Solve for

Explanation:

Recall that

With same base, we can write this equation:

By subtracting  on both sides,

### Example Question #1 : Exponents And Rational Numbers

Solve for .

Explanation:

Since  we can rewrite the expression.

With same base, let's set up an equation of .

By subtracting  on both sides, we get .

Take the square root of both sides we get BOTH  and

### Example Question #1 : Exponents And Rational Numbers

Solve for .

Explanation:

They don't have the same base, however: .

Then . You would multiply the  and the  instead of adding.

### Example Question #1 : How To Find An Exponent From A Rational Number

Solve for .

Explanation:

Method

They don't have the same bases however: . Then

You would multiply the  and the  instead of adding. We have

Divide  on both sides to get .

Method :

We can change the base from  to

This is the basic property of the product of power exponents.

We have the same base so basically

Solve for .