Review of Integer Exponents and Square Roots

Exponentiation is used to express iterated multiplication, much like multiplication is used to express iterated addition. Exponentiation is most naturally understood with integral exponents. We write $$a^{b} = a \times \ldots \times a$$ where some number $a$ is being multiplied by itself $b$ times. $a$ is called the $\mathbf{base}$ and $b$ is called the $\mathbf{exponent}$. For example, $$5^{3} = 5 \times 5 \times 5 = 125$$ Ready to practice?

Now we have a good understanding for exponentiation with positive integer exponents. What about if $b$ is not positive?

Consider how $a^{b}$ relates to $a^{b-1}$.

$a^{b-1}$ is $b-1$ $a$'s being multiplied together. Then $a^{b}$ is $a$ times $a^{b-1}$.

Let us consider when $b=0$. Note that $a^{1}=a$ for all numbers $a$.

What about negative exponents? If decreasing an exponent by 1 corresponds to dividing the answer by the base, we can develop an understanding for negative exponents by working from $a^{0}=1$.

Note that $7^{-2}=\frac{1}{7^{2}}$.

In fact, through the following exploration we have discovered a powerful result, namely

$$a^{-b}=\frac{1}{a^{b}}$$

which allows us to transition easily between positive and negative exponents.

At this point you know enough about exponentials to prove this useful tool by thinking about it with a general base and a general exponent. Just note that throughout this segment we assumed that b was an integer (though the result holds for any b, as we might expect).

Now continue to learn about rational exponents.

Consider evaluating the exponent $a^b$ as follows for a positive integer $b$: multiply the integer $a$ by itself $b$ times. For instance:

$2^2 = 2 \times 2 = 4$

$2^3 = 2 \times 2 \times 2 = 8$

This works regardless of the exponent as long as it is a positive integer. If the numbers get too big to feasibly do computations in your head, in some cases you will have access to a calculator. Without it, sometimes some tricks are needed. For example, if you wish to calculate $2^6$, expand the exponent as follows:

$2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) =$ $8 \times 8 = 8^2 = 64$

This is always a good skill to practice. Before long these can be done mentally with progressively more ease.

If the exponent is a negative integer, change the sign of the exponent and invert the number. In terms of an equation:

$a^{-b} = \frac{1}{a^b}$

Then simplify the fraction using the techniques described earlier. It can be seen with this logic that $2^{-6} = \frac{1}{64}$.

What is an Exponent?

An exponent is the power to which a value is to be raised. For example, in the term $2^3$, $3$ is considered the exponent that $2$ is being raised to. This would be written out like: $2 \times 2 \times 2$, where $2$ is being multiplied by itself $3$ times.

Here is a table to remind you of the exponential rules

What is a Square Root?

A square root is the fractional power to which a value is to be raised. If a value is raised to the $\frac{1}{3}$ power, then this is the equivalent of taking the cube root of the number. Therefore, raising a number to the $\frac{1}{2}$ power is the equivalent of taking the square root.

If you want more in depth descriptions, or a broader review, you can click here!

Basic Properties of Exponents

When you have a number of the form $a^b$ the $b$ is the exponent and that means that $a$ will be multiplied $b$ times to evaluate $a^b$. Now there are many special properties of exponents.

First of all we can observe that $a^0 = 1$ always. You are multiplying $a$, $0$ times to get $1$.

Let us try a problem!

$a^{-b} = \frac{1}{a^b}$. So for example, if we have $3^{-2}$ that would be $\frac{1}{3^2} = \frac{1}{9}$.

Let us try a problem!

$a^m \cdot a^n = a^{m+n}$. Let us see why this is true. If we have for example $a^5 \cdot a^3 = a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a$. That is $a$ multiplied eight times, which is $a^8$. We know that $5+3 = 8$

We can try a problem on this!

$\frac{a^m}{a^n} = a^{m-n}$. Let us see why this is true. If we have for example $a^5 \div a^3$, then that is $\frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a} = a \cdot a = a \cdot a$. That is $a$ multiplied two times which is $a^2$. We know that $5-3 = 2$

We can try a problem on this!

$(a^m)^n = a^{mn}$. Let us see why this is true. For example if we have $(a^5)^3$, we basically do $a^5 \cdot a^5 \cdot a^5$ which is $a^15$ and we know that $5 \cdot 3 = 15$.

I hope you learned and enjoyed from this post about the basic properties of exponents!