# SAT Math : How to find patterns in exponents

## 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?

-4

-9

3

6

-2

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 #2 : 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?

3p

p

(3/2)p

(2/3)p

2p

(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 #3 : How To Find Patterns In Exponents

Simplify 272/3.

729

125

9

3

27

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 #4 : How To Find Patterns In 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 #5 : How To Find Patterns In Exponents

If and , then what is ?

Explanation:

We use two properties of logarithms:

So

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

Evaluate:

Explanation:

, here  and , hence .

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

Solve for

None of the above

Explanation:

=

which means

### Example Question #8 : 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 #9 : How To Find Patterns In Exponents

Write in exponential form:

Explanation:

we get

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

Write in exponential form: