### All SAT Math Resources

## Example Questions

### Example Question #1 : How To Find Patterns In Exponents

If a^{x}·a^{4} = a^{12} and (b^{y})^{3} = b^{15}, what is the value of x - y?

**Possible Answers:**

-4

-9

3

6

-2

**Correct answer:**

3

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.

### Example Question #2 : How To Find Patterns In Exponents

If p and q are positive integrers and 27^{p }= 9^{q}, then what is the value of q in terms of p?

**Possible Answers:**

3p

p

(3/2)p

(2/3)p

2p

**Correct answer:**

(3/2)p

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 3^{3p }= 3^{2q}. So then 3p = 2q, and q = (3/2)p is our answer.

### Example Question #3 : How To Find Patterns In Exponents

Simplify 27^{2/3}.

**Possible Answers:**

729

125

9

3

27

**Correct answer:**

9

27^{2/3} is 27 squared and cube-rooted. We want to pick the easier operation first. Here that is the cube root. To see that, try both operations.

27^{2/3} = (27^{2})^{1/3} = 729^{1/3} OR

27^{2/3} = (27^{1/3})^{2} = 3^{2}

Obviously 3^{2} is much easier. Either 3^{2} or 729^{1/3} will give us the correct answer of 9, but with 3^{2} it is readily apparent.

### Example Question #4 : How To Find Patterns In Exponents

If and are integers and

what is the value of ?^{ }

**Possible Answers:**

**Correct answer:**

To solve this problem, we will have to take the log of both sides to bring down our exponents. By doing this, we will get .

To solve for we will have to divide both sides of our equation by to get .

will give you the answer of –3.

### Example Question #5 : How To Find Patterns In Exponents

If and , then what is ?

**Possible Answers:**

**Correct answer:**

We use two properties of logarithms:

So

### Example Question #6 : How To Find Patterns In Exponents

Evaluate:

**Possible Answers:**

**Correct answer:**

, here and , hence .

### Example Question #7 : How To Find Patterns In Exponents

Solve for

**Possible Answers:**

None of the above

**Correct answer:**

=

which means

### Example Question #8 : How To Find Patterns In Exponents

Which of the following statements is the same as:

**Possible Answers:**

**Correct answer:**

Remember the laws of exponents. In particular, when the base is nonzero:

An effective way to compare these statements, is to convert them all into exponents with base 2. The original statement becomes:

This is identical to statement I. Now consider statement II:

Therefore, statement II is not identical to the original statement. Finally, consider statement III:

which is also identical to the original statement. As a result, only I and III are the same as the original statement.

### Example Question #9 : How To Find Patterns In Exponents

Write in exponential form:

**Possible Answers:**

**Correct answer:**

Using properties of radicals e.g.,

we get

### Example Question #10 : How To Find Patterns In Exponents

Write in exponential form:

**Possible Answers:**

**Correct answer:**

Properties of Radicals