# Algebra II : Distributing Exponents (Power Rule)

## Example Questions

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### Example Question #1 : Distributing Exponents (Power Rule)

What is the largest positive integer, , such that  is a factor of ?

16

5

20

10

8

16

Explanation:

. Thus,  is equal to 16.

### Example Question #2 : Distributing Exponents (Power Rule)

Simplify the expression:

Cannot be simplified

Explanation:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

### Example Question #3 : Distributing Exponents (Power Rule)

Order the following from least to greatest:

Explanation:

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent.  Then they can be easily compared.

, , and .

Thus, ordering from least to greatest: .

### Example Question #4 : Distributing Exponents (Power Rule)

Simplify:

Explanation:

Step 1: Distribute the exponents in the numberator.

Step 2: Represent the negative exponents in the demoninator.

Step 3: Simplify by combining terms.

### Example Question #5 : Distributing Exponents (Power Rule)

Simplify:

Explanation:

Use the power rule to distribute the exponent:

### Example Question #6 : Distributing Exponents (Power Rule)

Simplify:

Explanation:

Step 1: Distribute the exponent through the terms in parentheses:

Step 2: Use the division of exponents rule.  Subtract the exponents in the numerator from the exponents in the denominator:

### Example Question #7 : Distributing Exponents (Power Rule)

Simplify.

Explanation:

When a power applies to an exponent, it acts as a multiplier, so 2a becomes 4a and -b becomes -2b. The negative exponent is moved to the denominator.

### Example Question #8 : Distributing Exponents (Power Rule)

Simplify .

Explanation:

When faced with a problem that has an exponent raised to another exponent, the powers are multiplied:  then simplify: .

### Example Question #9 : Distributing Exponents (Power Rule)

Solve:

Explanation:

Solve each term separately.  A number to the zeroth power is equal to 1, but be careful to apply the signs after the terms have been simplified.

### Example Question #10 : Distributing Exponents (Power Rule)

Simplify this expression:

Explanation:

is the correct answer because the order of operations were followed and the multiplication and power rules of exponents were obeyed. These rules are as follows: PEMDAS (parentheses,exponents, multiplication, division, addition, subtraction), for multiplication of exponents follow the format , and .

First we simplify terms within the parenthesis because of the order of operations and the multiplication rule of exponents:

Next we use the power rule to distribute the outer power:

=

**note that in the first step it isn't necessary to combine the two x powers because the individuals terms will still add to x^16 at the end if you use the power rule correctly. However, following the order of operations is a great way to avoid simple math errors and is relevant in many problems.

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