# SAT Math : How to find patterns in exponents

## Example Questions

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

What digit appears in the units place when is multiplied out?

Explanation:

This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.

Observe the first few powers of 2:

2= 2, 2= 4, 2= 8, 2= 16, 2= 32, 2= 64, 2= 128, 2= 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.

The second number in the sequence is 4, so the answer is 4.

### Example Question #21 : Gre Quantitative Reasoning

Which of the following is a multiple of ?

Explanation:

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least  in the prime factorization of the given number.  For each of the answer choices, you have:

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

Simplify the following:

Explanation:

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the  into its prime factors:

Note that these have a common factor of . Therefore, you can rewrite the numerator as:

Now, put this back into your fraction:

### Example Question #21 : Pattern Behaviors In Exponents

.

Express  in terms of .

Explanation:

### Example Question #271 : Exponents

Express  in terms of .

Explanation:

### Example Question #272 : Exponents

Express  in terms of .

Explanation:

### Example Question #273 : Exponents

Express  in terms of .

Explanation:

To solve this problem, first convert all numbers to have the like base of five.

Recall the rule of exponents which states, when multiplying like bases we add their exponents.

Since we have the same base we can set the exponents equal to each other and solve.

### Example Question #685 : Algebra

Solve for :

This statement has no solution.

This statement has no solution.

Explanation:

To solve this problem, first convert all numbers to have the like base of two

Recall that when exponents are raised to another power the exponents are multiplied together.

Since we have the same base we can set the exponents equal to each other and solve.

This statement is identically false, so the original statement has no solution.

### Example Question #271 : Exponents

Solve for :

The equation has no solution.

Explanation:

To solve this problem, first convert all numbers to have the like base of three.

Since we have the same base we can set the exponents equal to each other and solve.

### Example Question #281 : Exponents

What is the unit's digit of ?

Explanation:

Notice that there is a repeating pattern in the unit's digit of the powers of 2:

and so on. The pattern is four numbers long, which means we need to count how many complete blocks of 4 numbers are in the 357-number pattern. That is,

with a remainder of 1.

The remainder is all we need, since it tells us where in the pattern we need to be: the first place, with a unit's digit of 2.

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