### All GRE Math Resources

## Example Questions

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

find x

8^{x}=2^{x+6}

**Possible Answers:**

4

3

2

2 or -1

-1

**Correct answer:**

3

8 = 2^{3}

(2^{3})^{x} = 2^{3x}

2^{3x} = 2^{x+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 #2 : Exponents And Rational Numbers

Compare and .

**Possible Answers:**

The relationship cannot be determined from the information given.

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

can be written as

Since there is a common base of , we can say

or .

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

Solve for .

**Possible Answers:**

**Correct answer:**

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 #5 : Exponents And Rational Numbers

Solve for .

**Possible Answers:**

**Correct answer:**

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 #2 : Exponents And Rational Numbers

Solve for

**Possible Answers:**

**Correct answer:**

Recall that .

With same base, we can write this equation:

.

By subtracting on both sides, .

### Example Question #3 : Exponents And Rational Numbers

Solve for .

**Possible Answers:**

**Correct answer:**

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 #8 : Exponents And Rational Numbers

Solve for .

**Possible Answers:**

**Correct answer:**

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

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

.

### Example Question #9 : Exponents And Rational Numbers

Solve for .

**Possible Answers:**

**Correct answer:**

There are two ways to go about this.

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 .

### Example Question #10 : Exponents And Rational Numbers

Solve for .

**Possible Answers:**

**Correct answer:**

Since we can write .

With same base we can set up an equation of

Divide both sides by and we get .

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