## Example Questions

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

If ax·a4 = a12 and (by)3 = b15, what is the value of x - y?

-2

-9

3

6

-4

3

Explanation:

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 #1 : How To Find Patterns In Exponents

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

p

2p

(3/2)p

(2/3)p

3p

(3/2)p

Explanation:

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 33p = 32q. So then 3p = 2q, and q = (3/2)p is our answer.

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

Simplify 272/3.

27

9

125

729

3

9

Explanation:

272/3 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.

272/3 = (272)1/3 = 7291/3 OR

272/3 = (271/3)2 = 32

Obviously 32 is much easier. Either 32 or 7291/3 will give us the correct answer of 9, but with 32 it is readily apparent.

### Example Question #252 : Exponents

If and are integers and what is the value of ?      Explanation:

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 #1 : How To Find Patterns In Exponents

If and , then what is ?      Explanation:

We use two properties of logarithms:  So ### Example Question #2 : How To Find Patterns In Exponents

Evaluate:       Explanation: , here and , hence .

### Example Question #251 : Exponents

Solve for   None of the above    Explanation: =  which means ### Example Question #2 : How To Find Patterns In Exponents

Which of the following statements is the same as:        Explanation:

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 #134 : Exponents

Write in exponential form:       Explanation:

Using properties of radicals e.g., we get ### Example Question #3 : How To Find Patterns In Exponents

Write in exponential form:       Explanation:    