**Negative Exponents** *KS3 Revision*

## What you need to know

**Things to remember:**

- A negative power creates a reciprocal (A fraction with 1 on top)
- If we multiply with a negative power, we subtract it.
- If we divide with a negative power, we add it.

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 3

We have seen previously that we can write a negative exponent as the reciprocal with a positive power using two steps:

**Step 1: **Remove the negative symbol.

**Step 2: **Write as a reciprocal of the number and power.

Exponent **Step 1 Step 2**

5^{-3} 5^{3} \dfrac{1}{5^3}

6^{-7} 6^{7} \dfrac{1}{6^7}

2^{-9} 2^{9} \dfrac{1}{2^9}

But how can we use this to help us?

*Simplify the following by writing as a single power 2^5\times 2^{-3}*

* *

We’ll try and look at this using 5 steps.

**Step 1: **Rewrite the negative power as a reciprocal with a positive power.

2^{-3}=\frac{1}{2^3}

2^5\times2^{-3}=2^5\times\frac{1}{2^3}

**Step 2:** Write as a single fraction.

*Hint: **2^5 is just a number, and when we multiply a fraction by a number, all we need to do is multiply the numerator (the top) by it.*

* *

2^5\times\frac{1}{2^3}=\frac{2^5}{2^3}

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**Step 3: **Write the powers as actual numbers.

\frac{2^5}{2^3}=\frac{32}{8}

**Step 3: **Simplify the fraction.

*Hint: **We can simplify this fraction by turning it into a division.*

* *

\frac{32}{8}=32\div8=4

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**Step 4:** Write the number in **Step 3** as a power with the same base number as we started with.

4=2^2

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**Step 5: **Make this equal to the original question.

2^5\times2^{-3}=2^2

It is important to notice the relationship between the powers, 5-3=2. So, when we multiply with a negative power, we subtract it!

4^3\times4^{-1}=4^{3-1}=4^2

11^5\times11^{-9}=11^{5-9}=11^{-4}

a^9\times a^{-6}=a^{9-6}=9^3

b^{4a}\times b^{-2a}=b^{4a-2a}=b^{2a}

* *

*Simplify the following by writing as a single power 2^5\div2^{-3}*

Let’s try this by following some similar steps as before.

**Step 1: **Rewrite the negative power as a reciprocal with a positive power.

2^{-3}=\frac{1}{2^3}

2^5\div2^{-3}=2^5\div\dfrac{1}{2^3}

**Step 2: **Write the powers as actual numbers.

2^5\div\frac{1}{2^3} =32\div\frac{1}{8}

**Step 3: **Perform the division.

*Hint:** A division is looking for how many of the second number fit into the first. Start by thinking about how many \dfrac{1}{8} fit into 1.*

* *

There are 8 lots of \dfrac{1}{8} in 1.

There are 32 lots of 1 in 32.

There are 32 lots of 8 lots of \dfrac{1}{8}.

There are 32\times8=256

32\div\frac{1}{8}=256

* *

* *

**Step 4:** Write the number in **Step 3** as a power with the same base number as we started with.

256=2^8

* *

**Step 5: **Make this equal to the original question.

2^5\div2^{-3}=2^8

It is important to notice the relationship between the powers, 5+3=8. So, when we divide with a negative power, we add it!

4^3\div4^{-1}=4^{3+1}=4^4

11^5\div11^{-9}=11^{5+9}=11^{14}

a^9\div a^{-6}=a^{9+6}=9^{15}

b^{4a}\div b^{-2a}=b^{4a+2a}=b^{6a}

## Example Questions

**Question 1:** *Simplify the following by writing as a single power a^{13}\times a^{-4}*

a^{13}\times a^{-4}=a^{13-4}=a^9

**Question 2:** *Simplify the following by writing as a single power b^7\div b^{-7}*

b^7\div b^{-7}=b^{7+7}=b^{14}