## Algebra 1 Assignment Simplify Each Expression Negative Exponents

In algebra simplifying expressions is an important part of the process. Use this lesson to help you simplify algebraic expressions.

### Simplifying expressions

Simplifying an expression is just another way to say **solving a math problem**. When you **simplify** an expression, you're basically trying to write it in the **simplest** way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression:

4 + 6 + 5

If you **simplified** it by combining the terms until there was nothing left to do, the expression would look like this:

15

In other words, 15 is the **simplest** way to write 4 + 6 + 5. Both versions of the expression equal the exact same amount; one is just much shorter.

Simplifying **algebraic expressions** is the same idea, except you have variables (or letters) in your expression. Basically, you're turning a long expression into something you can easily make sense of. So an expression like this...

(13x + -3x) / 2

...could be simplified like this:

5x

If this seems like a big leap, don't worry! All you need to simplify most expressions is basic arithmetic -- addition, subtraction, multiplication, and division -- and the order of operations.

#### The order of operations

Like with any problem, you'll need to follow the** order of operations **when simplifying an algebraic expression. The order of operations is a rule that tells you the correct **order** for performing calculations. According to the order of operations, you should solve the problem in this order:

- Parentheses
- Exponents
- Multiplication and division
- Addition and subtraction

Let's look at a problem to see how this works.

In this equation, you'd start by simplifying the part of the expression in **parentheses**: 24 - 20.

2 ⋅ (24 - 20)^{2} + 18 / 6 - 30

**24** minus **20** is 4. According to the order of operations, next we'll simplify any **exponents**. There's one exponent in this equation: 4^{2}, or **four to the second power**.

2 ⋅ 4^{2} + 18 / 6 - 30

**4 ^{2}**is 16. Next, we need to take care of the

**multiplication**and

**division**. We'll do those from left to right: 2 ⋅ 16 and 18 / 6.

2 ⋅ 16 + 18 / 6 - 30

**2 ⋅ 16** is 32, and **18 / 6** is 3. All that's left is the last step in the order of operations: **addition** and **subtraction**.

32 + 3 - 30

**32 + 3** is 35, and **35 - 30** is 5.** Our expression has been simplified—there's nothing left to do.**

5

That's all it takes! Remember, you **must** follow the order of operations when you're performing calculations—otherwise, you may not get the correct answer.

Still a little confused or need more practice? We wrote an entire lesson on the order of operations. You can check it out here.

#### Adding like variables

To add variables that are the same, you can simply **add the coefficients**. So** 3 x + 6x** is equal to 9

*x*. Subtraction works the same way, so

**5**1

*y*- 4*y*=*y*, or just

*y*.

5y - 4y = 1y

You can also **multiply** and **divide** variables with coefficients. To multiply variables with coefficients, first multiply the coefficients, then write the variables next to each other. So **3 x ⋅ 4y** is 12

*xy*.

3x ⋅ 4y = 12xy

#### The Distributive Property

Sometimes when simplifying expressions, you might see something like this:

3(x+7)-5

Normally with the Order of Operations, we would simplify what is **inside** the parentheses first. In this case, however, x+7 can't be simplified since we can't add a variable and a number. So what's our first step?

As you might remember, the 3 on the outside of the parentheses means that we need to multiply everything **inside** the parentheses by 3. There are **two** things inside the parentheses: **x** and **7**. We'll need to multiply them **both** by 3.

3(x) + 3(7) - 5

3 · x is **3x** and 3 · 7 is **21**. We can rewrite the expression as:

3x + 21 - 5

Next, we can simplify the subtraction 21 - 5. 21 - 5 is **16**.

3x + 16

Since it's impossible to add variables and numbers, we can't simplify this expression any further. Our answer is **3x + 16**. In other words, 3(x+7) - 5 = 3x+16.

### Assessment

Want even more practice? Try out a short assessment to test your skills by clicking the link below:

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## Simplifying Expressions with Negative Exponents

#### Simplify the following expression:

The negative exponents tell me to move the bases, so:

Then I cancel as usual, and get:

When working with exponents, you're dealing with multiplication. Since order doesn't matter for multiplication, you will often find that you and a friend (or you and the teacher) have worked out the same problem with completely different steps, but have gotten the same answer in the end.

This is to be expected. As long as you do each step correctly, you should get the correct answers. Don't worry if your solution doesn't look anything like your friend's; as long as you both got the right answer, you probably both did it "the right way".

#### Simplify the following expression: (–3

*x*^{–1}*y*^{2})^{2}

I can proceed in either of two ways. I can either take care of the squaring outside, and then simplify inside; or else I can simplify inside, and then take the square through. Either way, I'll get the same answer. To prove this, I'll show both ways.

** simplifying first: **

** squaring first: **

Either way, my answer is the same:

#### Simplify the following expression: (–5

*x*^{–2}*y*)(–2*x*^{–3}*y*^{2})

Again, I can work either of two ways: multiply first and then handle the negative exponents, or else handle the exponents and then multiply the resulting fractions. I'll show both ways.

Either way, my answer is the same:

Neither solution method above is "better" or "worse" than the other. The way you work the problem will be a matter of taste or happenstance, so just do whatever works better for you.

#### Simplify the following expression:

The negative exponent is only on the *x*, not on the 2, so I only move the variable:

#### Simplify the following expression:

The "minus" on the 2 says to move the variable; the "minus" on the 6 says that the 6 is negative. These two "minus" signs mean *entirely* different things, and should *not* be confused.

I have to move the variable; I should not move the 6.

Affiliate

#### Simplify the following expression:

I'll move the one variable with a negative exponent, cancel off the *y*'s, and simplify:

URL: http://www.purplemath.com/modules/simpexpo2.htm

## Purplemath

Recall that negative exponents indicates that we need to move the base to the other side of the fraction line. For example:

(The "1's" in the simplifications above are for clarity's sake, in case it's been a while since you last worked with negative powers. One doesn't usually include them in one's work.)

In the context of simplifying with exponents, negative exponents can create extra steps in the simplification process. For instance:

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