## 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 $(-7)^3$

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.

$$7^3=343$$

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

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

## Example Questions

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 $(-3)^5=-243$

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 $(-2)^8=256$