An exponent is a power number.

For example, you might see 2³ (you may also see this written as 2^3). The top or last number is how many times you would multiply the first number by itself.

So, this is 2 x 2 x 2 = 8. And that's 2 multiplied by itself for 3 times. This is very different from multiplying 2 by the number three which gives you: 2 x 3 = 6.

“3” is an exponent because it is a little number at the top that tells us how many times to multiply the first number by itself.

They’re the basic rules you need to know for now.

You’ll learn more as we see it in action.

Let’s look at some questions now.

(x²)³(x⁶/x⁴) + 3x⁽¹²⁻⁴⁾ = ?
a) 4x⁸
b) x⁴
c) 3x²
d) x⁸

To answer this question, we need to apply some rules to make things easier. These rules are known as the index laws. The index laws you need to know are:

• (xⁿ)(xⁱ) = x⁽ⁿ⁺ⁱ⁾
• (xⁿ) / (xⁱ) = x⁽ⁿ⁻ⁱ⁾
• (xⁿ)ⁱ = xⁿⁱ

To begin with, we need to simplify the term (x²)³. Since this is an index to the power of another index, we can multiply the two indices (2 x 3 = 6) to get a result of x⁶.

Next, we need to simplify the term (x⁶/x⁴). Since this is an index divided by another index, we need to subtract the second index (6 - 4 = 2) to get a result of x².

Our expression now reads (x⁶)(x²) + 3x⁽¹²⁻⁴⁾. The first two terms are multiplied together, meaning we can add their indices to get a product of x⁽⁶⁺²⁾ or x⁸. Then, we can simplify the index of the second term 12 - 4 to get 8, meaning that our expression becomes x⁸ + 3x⁸. Since both terms are to the same power of x, we can now add them as normal to get 1 + 3(x⁸) = 4x⁸ (a) - Remember if it doesn't have a number in front of the x or any other letter, it is 1, so x⁸ is that same as 1x⁸.

Let's look at another question:

4 x 2 x 8² x 2 x 4² = 2^?

a) 6
b) 12
c) 14
d) 20

All of the terms in this equation (2, 4, and 8) are powers of base 2. In order to solve this question we need to get all terms to base 2.

First, we can consider that 4 = 2² and 8 = 2³. If we substitute this into the equation we then get:

2² x 2 x (2³)² x 2 x (2²)² = 2^?

Using the index law (xⁿ)ⁱ = xⁿⁱ so we know that this simplifies to:

2² x 2 x 2⁶ x 2 x 2⁴ = 2^?

Next, we use the index law (xⁿ)(xⁱ) = x⁽ⁿ⁺ⁱ⁾ to further simplify this equation.

Under this rule, 2⁽²⁺¹⁺⁶⁺¹⁺⁴⁾ = 2^?, meaning that ? = 2 + 1 + 6 + 1 + 4.

Adding up these numbers, we find that ? = 14. Therefore 4 x 2 x 8² x 2 x 4² = 2¹⁴.

One less effective way you might have tried to solve this question is by expanding the equation - that is to say, multiplying 4 by 2 by 8² and so on to get a result of 16384.

While this does eventually find you the answer, it means working with very large numbers, and it’s almost impossible to work out that 16384 = 2¹⁴ in your head.

It’s much easier to work with base 2 and indexes.

Key rules to remember

• Use index laws to simplify equation.
• If possible, get all values of x to the same power then simplify.
• Put all values in the same base to deal with larger numbers more easily (e.g. 4 = 2²).

Practice time!

Now, it's your turn to practice.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has two solutions videos after you complete the question. The first is a quick 60 second video that shows you how our expert answers the question quickly. The second video is a more in-depth 5-steps or less explainer video that shows you the steps to take to answer the question. It's really important that you review the second video because that's where you'll learn additional tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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