Exponents are a shorthand way of telling us that a number is multiplied by itself. The number that is multiplied by itself is called the base. The base is the larger number on the left. The smaller number on the right tells us how many times the number is multiplied by itself. It is called the exponent, or power.

We will usually read it as (base) to the power of (exponent), OR (base) to the (exponent) power.

For example, in the expression $4^3$

4 is the base, while 3 is the exponent. The exponent tells us the number of times the base is to be multiplied by itself. In our example, 4 (the base) is multiplied by itself 3 times (the exponent): $4\times4\times4$ We can call this 4 to the power of 3, or 4 to the third power.

Extra: Since the second and third powers are so common, we have special, short names for them - squared and cubed.

$4^2$ can be called simply 4 squared.

$4^3$ can be called simply 4 cubed.

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Exponents, or powers, are a shorthand way of writing that a number (the base) is multiplied by itself a certain amount of times (exponent). The exponent itself can be any number. For now, we'll focus on positive, whole numbers.

For example:

$4^2$ Is called: 'four to the second power' OR 'four to the power of 2' (or 'four squared'). The 4 will be multiplied by another 4 (two fours). $4^2=4 \times 4=16$

$4^3$ Is called: 'four to the third power' OR 'four to the power of 3' (or 'four cubed'). The 4 will be multiplied three times (three fours). $4^3=4 \times 4 \times 4=64$

Remember: The number that is multiplied by itself is called the base, and the number of times the number is multiplied is called the exponent.

Therefore, in the expression $4^2$

4 is the base, while 2 is the exponent. In this case the number 4 is multiplied by itself 2 times, therefore this expression will be called '4 to the second power' OR '4 to the power of 2' OR '4 squared.'

And in the expression $4^3$

4 is the base, while 3 is the exponent. In this case the number 4 is multiplied by itself 3 times, therefore this expression will be called '4 to the third power' OR '4 to the power of 3' or '4 cubed.'

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We learned in the Order of Operations that we first solve what's inside the parentheses and then continue with the rest of the equation. What comes next? Let's look at our order of operations acronym: PEMDAS.

The E in PEMDAS is for exponents! They come second and we will solve them after the parentheses and before going on to solve multiplication and division.

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Example exercises using exponents

Exercise 1

Task:

What exponent will make the following equation true?

$7^{\square}=49$

Solution:

We can find the answer using two different approaches.

Method 1 - Replacement:

We can try replacing the unknown exponent with a number, let's say $2$, and check if it fits the equation. In our case it seems that we have arrived at the correct answer.

We multiply each of the terms in parentheses by its power.

$9^4\times a^4\times x^4+4^{a\times x}=$

$9^4a^4x^4+4^{\left\{ax\right\}}$

Answer:

$9^4a^4x^4+4^{\left\{ax\right\}}$

Review questions

What is an exponent?

An exponent, or power, is a simple way to say that a number is multiplied by itself. A power has two elements: a base and an exponent. The exponent tells us the number of times the base is going to be multiplied by itself.

Let's see some examples:

Example 1

$3^4=$

Here the base is the number $3$ and the $4$ is the exponent, which means that the number $3$ must be multiplied $4$ times. Then we have the following:

The five must be multiplied by itself three times, so

$5^3=5\times5\times5=125$

Result

$5^3=125$

Why would I use an exponent?

An exponent can help us to simplify the multiplication of the same number. It is a simple way of indicating the number of times that number should be multiplied by itself.

The power of 0: Any number raised to the power of 0 is $1$.

$A^0=1$

2. The power of 1: Any number raised to the power of 1 will be the same number.

$A^1=A$

3. Multiplying powers with the same base:

$A^m\times A^n=A^{m+n}$

4. Dividing powers with the same base:

$\frac{A^m}{A^n}=A^{m-n}$

5. Multiplying powers with the same exponent:

$\left(A\times B\right)^n=A^n\times B^n$

6. Dividing powers with the same exponent:

$\left(\frac{A}{B}\right)^n=\frac{A^n}{B^n}$

7. Power of a power:

$\left(A^m\right)^n=A^{m\times n}$

8. Negative power:

$A^{-m}=\frac{1}{A^m}$

Using the properties

In order to understand when to use the different properties of exponents, we will need to understand the function of the properties themselves. Let's take a look at some examples:

Example 1

Task:

$7^5\times7^3=$

To solve this problem we will use the third property of exponents: multiplying powers with the same base:

To solve, we will use the fourth property of powers: dividing powers with the same base:

$\frac{8^6}{8^4}=8^{6-4}=8^2$

If we want to simplify the power we will get:

$8^2=8\times8=64$

Result

$\frac{8^6}{8^4}=8^2$

Example 3

Task:

Solve $\left(2^5\right)^3\times2^{-3}=$

In the first part we will use the seventh property of powers: power of a power. In the second part we will use the eigth property powers: negative powers.