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Whole Number Exponents

When you raise a number (called the base) to a whole number exponent (called the power), you're essentially multiplying that number by itself however many times the exponent says. Here's how it works:

Here's another example: $9^4$can also be written as: $$9^4 = 9 \cdot 9 \cdot 9 \cdot 9 = 6{,}561$$

The exponent might be any natural number. If the exponent is $1$, you can remove the exponent. It is the same as the identity property of multiplication.

$$6^1 = 6$$

To find $5^3,$ all we have to do is multiply $5$ with itself $3$ times: $$5 \cdot 5 \cdot 5 = 125$$

You can also raise expressions to whole powers. If you have an expression under an exponent (let's call it $n$), to evaluate the whole thing you'll have to multiply the expression by itself $n$ times: $$(2x + 5)^3 = (2x + 5) \cdot (2x + 5) \cdot (2x + 5).$$

What are Exponents?

Let's say you have one number (called a base), and to the top right of it, there's another number (called an exponent or a power). For example, consider: $$2^3$$

You can describe this expression by saying two raised to the power of three, two to the power of three, or two to the third power.

But what does this mean?

So to evaluate these types of expressions, multiply the base by itself a total of the value of the exponent times.

Here's another example: $$\begin{align} &\phantom{} \phantom{} \tiny{1} \phantom{12345} \tiny{2} \phantom{12553} \tiny{3}\\ 5^3 = &5 \times 5 \times 5 = 125 \end{align}$$

Let's do another example.

First, let's determine which number is the base and which number is the exponent.

Since $3$ is the number being raised to a power, $3$ is the base.

Since $4$ is the power that $3$ is being raised to, $4$ is the exponent. The exponent is the number to the top right of the base.

Now, let's evaluate $3^4.$ $$\begin{align} 3^4 &= 3 \text{ multiplied by itself } 4 \text{ times}\\ &=3 \times 3 \times 3 \times 3\\ &=(3 \times 3) \times (3 \times 3)\\ &= 9 \times 9\\ &= \boxed{81} \end{align}$$

(Video) Intro to Exponents

by Math Antics

Here's a good video by mathantics that explains the basics of exponents!

$\large{\text{Summary}}$

The video draws an analogy to the relationship between addition and multiplication.

$$2\times 3=6\\ \swarrow \downarrow\searrow\\ \quad \,\,\,2+2+2=6$$

We can think of multiplication is repeated addition. In the example above, the multiplication of $2$ by $3$ is representing the addition of $2$ to itself $3$ times. A number multiplied by another number can also be added repeatedly. That is, $2$ times $3$ means the addition of $3$ to itself $2$ times.

He then reviews the anatomy of an exponent: $$\text{base}\rightarrow 2^{3\,\leftarrow \text{ exponent/index}}$$ The base is the number we want to multiply, while the exponent is the number of copies of the base that we have.

$$\,\,\quad\quad 2^3=8\\ \swarrow \downarrow\searrow\\ \quad \,\,\,2\times 2\times 2=8$$

Similarly, exponentiation is repeated multiplication. In the example above, the exponentiation of $2$ by $3$ is representing the multiplication of $2$ by itself $3$ times total. Note that this is not equivalent to $3^2$, which is $3\times 3=9$, so exponentiation is not commutative. Sometimes we'll have negative exponents rather than whole number exponents. Sometimes the exponents will even be fractions! In fact, it can be any real number. But usually they're whole numbers.

Finally, some notation: a power of $2$ is often called "squared," while a power of three is often called "cubed."

$$5^4=5\times 5\times 5\times 5=625$$

$$4^3=4\times 4\times 4=64$$