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(Videos) Radicals

by mathman1024

Here's a video by mathman1024 explaining square roots and radicals.

$\large{\text{Summary}}$

The number $a$ is called a square root of $b$ if $a^2=b$. Sometimes $a$ will be a rational number, typically when it's an integer or a whole number. Otherwise it is usually an irrational number.

For example, we can say that $5$ and $-5$ are square roots of $25$. $$(5)^2=25\\ (-5)^2=25$$

This symbol $\sqrt{\ \ }$ is used for the principal square root, or the non-negative square root. If we were to write $\sqrt{25}$ this would only be $5$.

An example he looks at is: Evaluate $\sqrt{144}$. $$\sqrt{144}=12$$

If we want the negative part we just put a negative in front of the radical. $$-\sqrt{49}=-7$$

When taking a square root of a fraction, you just take the square root of both the numerator and denominator. If necessary, reduce the fraction afterward. $$\sqrt{\frac{4}{81}}=\frac{\sqrt{4}}{\sqrt{81}}=\frac29$$

So what happens if you have the minus sign inside the radical? $$\sqrt{-36}$$ This actually isn't a real number — it's an imaginary number, which you'll learn about later. For now you just have to know that you can't do anything with this. Just remember that: $$\sqrt{-36}\ne-6\\ \sqrt{-36}\ne-\sqrt{36}$$

Here's another video where he explains the $n^{\text{th}}$ root.

The number $a$ is called an $n^{\text{th}}$ root of $b$ if $a^n=b$. This is similar to square roots, only we use other numbers besides $2.$ We also write these a little differently. $$\sqrt[n]{\text{Expression}}$$ Here the $n$ is referred to as the index. This indicates what root it is we're looking for when we do root calculation. It could be the square root, cube root, fourth root, etc. The expression inside is called the radicand.

So an example would be find the cubed root of twenty-seven, $\sqrt[3]{27}$. You can restate the problem as, "What to the third power gives you $27?$" $$\sqrt[3]{27}\Longrightarrow(\ \ \ \ )^3=27$$ Here the answer would be $3$, because $3^3=27$. So, $$\sqrt[3]{27}=3$$

Try these next ones on your own and then watch the video to see the answers!

$\sqrt[4]{16}$

$\sqrt[3]{-125}$ (watch out, this answer is not a positive real number)

$\sqrt[5]{-1024}$ (watch out, this answer is not a positive real number)

Introduction to Square Roots

A square root can be thought of as the reverse of squaring a number. If we square a number $x,$ we get $x^2,$ and then taking the square root again gives $\sqrt{x^2} = x.$

But there's one minor warning sign I have to flash.

When we calculate square roots, we get two answers, one positive and one negative. If we're taking the square root of $x^2,$ then we get both $x$ and $-x$ as possible outputs, since both $x^2$ and $(-x)^2$ are equal to $x^2$ (unless, of course, $x^2 = 0,$ in which case $x = -x = 0$). We use the symbol $\sqrt{ }$ to represent the principal square root, or the non-negative square root.

Also, we can't obtain a real square root of a negative number, because we can't square any real number and get a negative number. Squaring always gives nonnegative numbers, so we make this rule so that squaring and square roots truly are inverse operations, with no cases where the "inverse" breaks down. Also, squaring is an example of introducing an exponent. In fact, squaring and taking a square root are considered opposite actions. They cancel each other out.

We can use this procedure to find square roots of numbers, as long as we understand how to square numbers.

We want to find a number $x$ such that $x^2 = 9.$ From guessing and checking, we can identify that $3^2 = 9,$ so $\sqrt{9} = 3.$ However, as discussed earlier, we must also use the negative of this value, meaning $-3$ is also a square root of $9$.

Try one more example for good measure.

We can handle this in the same way as the previous problem. We want to find a number $x$ such that $x^2 = 49.$ From guessing and checking we can identify that $7^2 = 49,$ so $\sqrt{49} = 7.$ However, as discussed earlier, we must also use the negative of this value. This means $-7$ is also a square root of $49$.

There are a couple of other things we can do with square roots. The first is to combine evaluating square roots with addition, subtraction, multiplication, and division to evaluate more complicated expressions. This is one of many examples of using the order of operations.

Example: Let's evaluate $\sqrt{6 + 3} - 2.$ Remember to take positive square roots.

How we do this is simplify everything inside the square root first. $6 + 3$ is inside a single square root, so

$$\sqrt{6 + 3} - 2 = \sqrt{9} - 2.$$

As discussed before, the value of the positive square root of $9$ is $3,$ so

$$\sqrt{9} - 2 = 3 - 2 = 1.$$

Now let's try one using multiplication and division.

Example: Let's evaluate $\sqrt{9} \times \sqrt{\frac{96}{6}}.$ Remember to take positive square roots

There is a division inside the second square root, so simplify that first.

$$\sqrt{9} \times \sqrt{\frac{96}{6}} = \sqrt{9} \times \sqrt{16}.$$

We know from before that $\sqrt{9} = 3,$ so

$$\sqrt{9} \times \sqrt{16} = 3 \times \sqrt{16}.$$

What's left is to find $\sqrt{16}.$ We know that $4^2 = 16,$ so $\sqrt{16} = 4,$ and

$$3 \times \sqrt{16} = 3 \times 4 = 12.$$

Now let's talk about a different type of problem: taking an expression that's in words and writing it in terms of symbols. This will help when trying to solve equations that are given to us in words.

Example: Write in symbols: add $3$ to the square root of a mystery number.

Typically we will call a "mystery number" $x,$ because that's the traditional name for a variable in algebra. Then we take the square root of the number, and then add $3$. This gives

$$\sqrt{x} + 3.$$

Notice we don't include the $+3$ under the square root. Why? Because we take the square root and then add $3,$ just as the directions told us. On the other hand, if we were to add $3$ and then take the square root, we'd have $\sqrt{x + 3}.$ Try plugging in different values of $x,$ and you'll see that (usually)

$$\sqrt{x} + 3 \neq \sqrt{x + 3}.$$

Finally, we need to talk about word problems. Since we're talking about squares and square roots, the problems of greatest interest will be those involving squares, as you might have expected. In particular, recall a special formula about shapes.

If $s$ is the length of a side of a square, and $A$ is the square's area, then

$$A = s^2.$$

In particular, taking the square root gives us $s = \pm \sqrt{A}.$ Normally this gives a positive and negative answer, but since $s$ is a side length, we discard the negative solution. For shapes, only the positive square root makes sense! Have you ever heard of something having negative length?

We're going to wrap up this explanation with a related example.

Percy's Dance has a square dance floor that has an area of $100$ square feet. Let's find the length of one side of the dance floor.

This just comes down to finding $\sqrt{100},$ or solving the equation

$$x^2 = 100.$$

Since we are dealing with a physical object, the dance floor, we want to only use the positive square root. Plug-and-chug different side lengths to notice that

$$10^2 = 100.$$

Thus our answer is $10$ feet (note how we remove the "squared" off of the units, since a side-length is a one-dimensional object).

That pretty much covers the basics of square roots. Now you can say you've dug up some of the roots of mathematics!!

(Video) Rational Square Roots

by Derek Owens

Here's a video by Derek Owens explaining what square roots are and how to solve them.

$\large{\text{Summary}}$

The square root of $b$ is a number $a$ such that $a^2=b.$

We know that $5^2=25.$ Also, $(-5)^2=25$. So, we can say the square roots of $25$ are $5$ and $-5$.

What is the square root of $16?$

What number, when multiplied by itself, gives us $16?$ The answers are $4$ and $-4.$

What is the square root of $100?$

What number, when multiplied by itself, gives $100?$ The answer is both $10$ and $-10.$

The radical symbol for square roots is $\sqrt{\phantom{0}}$. It is used to represent the unique non-negative square root of a number, also called the principal square root.

So, $\sqrt{81}$ equals $9$.

Negative numbers do not have real square roots. No real number multiplied by itself gives a negative number. $$\cancel{\sqrt{-36}}$$