**Evaluating Exponents with Negative Bases** *KS3 Revision*

## What you need to know

**Things to remember:**

- If the base number is negative and the power is odd, the answer will be negative.
- If the base number is negative and the power is even, the answer will be positive.

An exponent is a number written to the top right of a number and tells us how many times we multiply a number by itself.

6^2 = 6\times 6

6^3 = 6\times 6\times 6

6^4= 6\times 6\times 6\times 6

So, all we’re doing when we have exponents is lots of multiplications. When we do these multiplications, we refer to it as “evaluating the exponent”.

*Evaluate the following exponents:*

* *

5^2=5\times5=25

7^3=7\times7\times7=343

8^4=8\times8\times8\times8=4096

Let’s look at some examples with a base number of just -1, to keep things simple.

*Hint: **Remember, when we multiply two negative numbers, we get a positive answer, and when we multiply a negative number and positive number, we get a negative answer.*

(-1)^1=-1

(-1)^2=-1\times-1=1

(-1)^3=-1\times-1\times-1=-1

(-1)^4=-1\times-1\times-1\times-1=1

(-1)^5=-1\times-1\times-1\times-1\times-1=-1

(-1)^6=-1\times-1\times-1\times-1\times-1\times-1=1

There are two key points we should take away from these examples:

- When the base is negative and the power is odd, the answer is negative
- When the base is negative and the power is even, the answer is positive.

** **

*If we evaluate (-4)^3 will the answer be positive or negative?*

The answer will be negative, because the base number is negative, and the power is odd.

*If we evaluate (8)^7 will the answer be positive or negative?*

The answer will be positive, because the base number is positive.

*Hint: **The base number is positive, so the answer will always be positive.*

* *

*If we evaluate (-9)^2 will the answer be positive or negative?*

The answer will be positive, because the base number is negative, and the power is odd.

*Hint: **Whenever the power is even, the answer will be positive.*

*If we evaluate (-7)^7 will the answer be positive or negative?*

The answer will be negative, because the base number is negative, and the power is odd.

* *

*If we evaluate (-11)^{-4} will the answer be positive or negative?*

The answer will be positive, because the base number is negative, and the power is odd.

* *

*Hint: **It doesn’t matter that the power is negative, it is still even.*

So, we can actually do evaluate exponents with negative bases in three steps:

*Evaluate (-4)^4*

**Step 1: **Determine whether the answer will be positive or negative.

The base number is negative, and the power is even, so the answer will be positive.

**Step 2: **Evaluate the exponent without the negative.

4^4=256

**Step 3: **Write the answer appropriately, as positive or negative.

The answer will be positive, so (-4)^4=256

*Evaluate (-7)^3*

*S***tep 1: **Determine whether the answer will be positive or negative.

The base number is negative, and the power is odd, so the answer will be negative.

**Step 2: **Evaluate the exponent without the negative.

7^3=343

**Step 3: **Write the answer appropriately, as positive or negative.

The answer will be positive, so (-7)^3=-343

## Example Questions

**Question 1:** *Evaluate (-3)^5*

**Step 1: **Determine whether the answer will be positive or negative.

The base number is negative, and the power is odd, so the answer will be negative.

**Step 2: **Evaluate the exponent without the negative.

3^5=243

**Step 3: **Write the answer appropriately, as positive or negative.

The answer will be negative, so (-3)^5=-243

**Question 2:** *Evaluate (-2)^8*

**Step 1: **Determine whether the answer will be positive or negative.

The base number is negative, and the power is even, so the answer will be positive.

**Step 2: **Evaluate the exponent without the negative.

2^8=256

**Step 3: **Write the answer appropriately, as positive or negative.

The answer will be positive, so (-2)^8=256