# Common Core Standards: Math

### Expressions and Equations 8.EE.A.1

1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

We're guessing your students aren't huge fans of rules. They're in eighth grade. They're teenagers, and teenagers are automatically supposed to rebel against any rules. Just look at James Dean, the Fonz, and even Carly and Sam from iCarly. We get it.

The thing is, these rules are cool. They'll help your students deal with those pesky little numbers called exponents. That will help them get a high school diploma, which will help them get into college, which will help land them a great job, which will lead to them being cooler than Fonzie on a motorcycle.

But before they can do that, they should know how to handle exponents and it all starts with the List of Rules. Sounds official, doesn't it?

1. When multiplying terms with the same base, the exponents are added together. For instance, a3 · a5 = a8 because 3 + 5 = 8.
2. If multiplication means adding exponents, then dividing means subtracting exponents. That means n7 ÷ n3 = n4 because 7 – 3 = 4.
3. When there's an exponent on an exponent, the exponents are multiplied. For example, (x5)2 = x10 because 5 · 2 = 10.
4. A negative exponent means we take the reciprocal of the base. So 2-2 = ½2 = ¼.
5. Anything raised to the 0 power is 1. So x0 = 1 and 570 = 1 and 00 = 1. This one fools a lot of wannabe cool people.

Students should also know that these rules are meant to simplify their lives, not complicate them. After all, being cool is a way of life—and the last thing we want to do is cramp their style.

#### Drills

1. Simplify ¼3.

164

Raise both the numerator and denominator to the third power. We know that 13 = 1, and 43 = 64. So our answer is (C). All the other answers are incorrect because (A) results from multiplying 3 · 4 = 12, (B) results from adding 3 + 4 = 7, and (D) results from substituting the numerator with 3.

2. Simplify x3 · x6.

x9

To multiply powers of the same base, add the exponents. Since 3 + 6 = 9, the answer we're looking for is x9. The other answers come from incorrectly combining the exponents.

3. Simplify 5-2.

125

A negative exponent means we need to take the reciprocal of the base. In other words, we have ⅕2. Since 52 = 25, we know that (D) is the right answer. Option (A) results from multiplying the base by the exponent, (B) forgets to take the reciprocal, and (C) takes the reciprocal but multiplies instead of raising the base.

4. Simplify 17,4120.

1

Anything raised to the 0 power is 1, remember? Anything. Including 17,412. Now wipe that smirk off your face as you think of all those wannabes who thought the answer was zero.

5. Simplify (-1)5.

-1

Multiply -1 by itself 5 times. We'll still end up with some kind of 1, which means no fives or fifths involved. Four of the negatives will cancel, the last one won't. So we're left with -1 as our answer.

6. Simplify ⅔-2.

94

A negative exponent means we take the reciprocal of the base, which means turning ⅔ into 32. Next, we square it to get (D). It wouldn't make sense for our answer to be negative and (C) forgets to take the reciprocal.

7. Simplify 23 · 33.

216

If you're multiplying factors with two different bases, you simplify each one separately and then find the product. Since we can't combine the bases here, we just have to calculate them individually. 23 · 33 = 8 · 27 = 216. The other answers result from incorrectly combining the numbers.

8. Why does anything to the zero power equal one?

Because the when you divide something by itself, subtracting the exponents gives an exponent of zero

Dividing anything by itself gives an answer of 1, and it also gives you something to the zero power. In other words, x ÷ x = x1 ÷ x1 = x1 – 1 = x0 = 1.

9. Simplify (0.1)3.

0.0001

If it helps, change 0.1 to 110. To raise it to the third power, raise both the numerator and denominator to the third power.

10. Simplify (93 ÷ 35)2.

9