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(Video) Exponents and Square Roots

by Math Antics

Here's a video by mathantics that explains square roots and how they're related to exponents.

$\large{\text{Summary}}$

Recall that an inverse operation is an operation that undoes another. For example, subtraction undoes addition, division undoes multiplication, and vice versa. Similarly, a root undoes an exponent, but more specifically, the square root undoes a number being squared. All in all, once you undo something, you end up with the original number.

We can also say that a number being squared undoes a square root—it's a two-way street. And this is the ONLY operation that does that; we can see that inverse operations are unique. It's exactly why the additive inverse of a number is unique, for example.

For example, say we have $4^2$ and we solve it, we know that the answer is $16$. But what if we wanted to go backwards? Our discussion earlier suggests we can do that! If we start with the $16$, and want to get back to $4$, we use the root operation. $$\sqrt{16}=4$$

Here, the $\sqrt{\phantom{333}}$ symbol is called the root sign. It is also often referred to as a radical sign. The number you want to take the root of goes underneath the sign. $$\sqrt{16}$$ So, when you see a square root, this should indicate that you're looking for some number that can be multiplied with itself multiple times to equal $16$. How many times though? That depends on the power of the base. Whatever power the base is, that is how many times it must be multiplied with itself to get the number under the radical sign. How do we know what that number is? That's given to us by the root symbol. We actually haven't included it yet. It would look like: $$\sqrt[2]{16}$$ Now, we know that we need to find a number that's multiplied together two times to equal $16$. $$\sqrt[2]{16}=4$$

Now, let's change the little two into a little four. $$\sqrt[4]{16}$$ This problem is now asking us what number multiplied together four times, gives us $16$. The answer is $2$. $$\sqrt[4]{16}=2$$

Luckily, you'll usually only deal with numbers that have perfect roots. Most of them will be $$\sqrt[2]{\#}\text{ or }\sqrt[3]{\#}$$ These are called square roots and cubed roots respectively. They get special names because they're the most common. The square root is the most common of all the roots, so it gets a special treatment. When writing a square root, we don't include the $2$. The root symbol, $\sqrt{\ }$, with no number is always the square root. $$\sqrt[2]{\#}=\sqrt{\#}$$

Perfect Squares are the resultant of whole numbers being multiplied together. So, if you take the square root of a perfect square, you get a nice whole number as your answer, instead of a long scary decimal. Otherwise, you'll get a decimal, and one that goes on forever at that. These decimals are called nonterminating decimals. Moreover, if the square root of a number $a$ gives a nonterminating decimal, then $\sqrt{a}$ is called an irrational number. If this is not the case, then $\sqrt{a}$ is a rational number.

When dealing with roots it's also worth asking about zero. Well, zero is a very special number in many problems. But here what you need to know is that

$$\sqrt{0} = 0$$

Also,

$$0^2 = 0$$

It is a coincidence these operations spit out the number you started with. The second equation is true because of the zero property of multiplication.

What are Square Roots?

Say we have a number ${x}$. If we multiply this number by itself, we'll have: $$x\times x\text{, which equals }x^2$$ Now, say we want to find the square root of $x^2$. By definition, the square root of a number is the number which, when multiplied by itself, produces a product equal to the given number. Therefore, the square root of $x^2$ would be $x$.

But wait. What about $(-x)\times(-x)$? That would also be $x^2$! So, how do we know if $x$ or $-x$ is the square root of $x^2$?

The answer is both. There is a name for the non-negative square root of any given number, and that is the principal square root. So, if a question asks for the principal square root of a given number, the answer should be non-negative. Often times, this term will not be used directly but will be implied. Other times, it is necessary to give both the non-negative and negative versions of the answer.

Square roots can be represented with the radical symbol:

$$\sqrt[2]{n}$$

However, the $2$ is often omitted. Square roots are more often written as $$\sqrt{n}$$

It is also sometimes written as a fractional exponent: $n^{\frac{1}{2}}$. But we'll stick to $\sqrt{n}$ notation from now on.

Square Root of a Perfect Square

We are now going to take a look at some of the types of problems you might encounter about finding a square root of a perfect square.

Before you start these problems, make sure to go over the basics of finding a square root of a perfect square from the other explanations in this section.

Since $9^2 = 81$, $$\sqrt{81}=\sqrt{9^2}= 9$$

$36$ is the only perfect square in the list because it can be expressed as $6^2=36$.

Since $3^2 =9$, $$\sqrt{9} = \sqrt{3^2}= 3$$ Therefore, $\sqrt{9}$ is an integer.