## 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.

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

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.

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

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

$$4=2^2$$

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 $b^7\div b^{-7}$

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