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

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

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

Need some extra help? Find a Maths tutor now

Or, call 020 3633 5145