Roots

A square root is a partial inverse function to a power. Most often we work with the square root, which looks for a number that, when multiplied with itself, gives the original number we square rooted.

The square root

For the square root we use the character $\sqrt{}$, and to avoid having to write the argument of the square root in parentheses like this: $\sqrt{}$(25), we make a horizontal line above the whole argument (the expression we want to square root), like this: $\sqrt{25}$.

At the beginning, we will only be dealing with the square root of a real number. We would define it as follows:

$$\sqrt{a}\cdot\sqrt{a}=a$$

If you multiply the square root of a with the square root of a, then you get the number a. So for the number 9, the square root would be 3, because 3 · 3 = 9 is valid.

We can also write the previous expression as follows:

$$\left(\sqrt{a}\right)^2=a$$

It's just a different notation for the previous multiplication, namely using a power.

The important fact is that this equation is valid only for those x, which are from the definitional domain of the square root. This is because we cannot square root a negative number. We can square any positive number, we can square zero, but we can't square a negative number? Why can't we do it?

Suppose we want to calculate the square root of −25. We basically have two choices of what number to choose. Either positive or negative. If we choose positive, we get 5 · 5 = 25. We get 25, but we want −25. So we try to choose −5; only after multiplying, we get 25 again. In short, if you multiply two negative numbers, you get a positive number. Likewise, if you multiply two positive numbers. You would have to multiply −5 · 5 to get −25 and you can't do that, you are multiplying two different numbers, even if they differ only in sign. Therefore, you cannot calculate the square root of a negative number.

Multiple square roots

Just like we can multiply an expression to the second, to the third, to the fourth, we can have the third and fourth and finally n-th square root of a real number. This is usually written above the beak, like this:

$$\sqrt[5]{32}$$

This notation denotes the fifth root of the number thirty-two. n-th root is defined using n-th power as follows:

$$\sqrt[n]{a}=b\Leftrightarrow b^n=a;\qquad n\in\mathbb{N}, a,b\ge0$$

If we put two after n, we get the square root as we defined it a moment ago. For short, we could also define it like this:

$$\left(\sqrt[n]{a}\right)^n=a$$

The convention is that we don't need to write a two there for the square root, so these notations are equivalent:

$$\sqrt{64}=\sqrt[2]{64}$$

So, for example, for the fourth root, it would be:

$$\sqrt[4]{a}\cdot\sqrt[4]{a}\cdot\sqrt[4]{a}\cdot\sqrt[4]{a}=a$$

And a concrete example:

$$\sqrt[3]{64}=4$$

And the converse is true 4 · 4 · 4 = 64.

The square root of a negative number

We already know that there is no square root of a negative number, because a², where a is a real number, will never be negative. However, for multiple square roots, we no longer multiply only twice, so there may be cases where n-th square root of a negative number exists. So how is this?

It bothered us that when we multiply a negative number twice, we get a positive number. For the result −25 we needed the product −5 · 5. But what happens when we multiply three negative numbers? After multiplying the first two negative numbers, we have a positive number. But after multiplying by the last negative number, we get back a negative number. If we multiply one more time (four times total), we are back in the positive numbers.

The lesson here is that if we n-times multiply a negative number, when n is even, the resulting product is positive, when it is odd, the product is negative. We conclude that there is a n-th square root of a negative number when n is odd. Example:

$$\sqrt[3]{-216}=-6$$

Conversion to a power

We can easily convert square roots to powers, and often do, because it makes it easier to work with square roots. The following relationship holds:

$$\sqrt[n]{a}=a^{\frac{1}{n}}$$

If we have n-that square root of a, it is the same as if we multiplied a to 1/n. This can also come in handy when you need to enter a higher square root than the other one somewhere. Often you can't do this directly, but you can enter a fraction in the exponent. To have the graph of the fourth root plotted, have the graph of the function

$$f(x)=x^{1/4}.$$

The most commonly used is the square root conversion:

$$\sqrt{x}=x^{\frac12}$$

Formulas for square roots

Because a square root can be easily converted to a power, expressions with square roots inherit the formulas for common powers. So the following relationships apply:

$$\Large \sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}\qquad a,b\ge0$$

$$\Large \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\qquad a,b\ge0$$

$$\Large \sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$$

$$\Large \sqrt[n]{a^k}=a^{\frac{k}{n}}\qquad a>0$$

At the same time, the following basic relations hold:

$$\Large \sqrt[n]{0}=0$$

$$\Large \sqrt[n]{1}=1$$

$$\Large \sqrt[1]{a}=a$$

Partial square roots

Partial square root is a technique used to simplify square roots and uses one of the preceding formulas:

$$\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot\sqrt[n]{b}\qquad a,b\ge0$$

Namely, we can decompose some expressions below the square root into a product, then decompose that product into two square roots, and finally square one of those square roots. A typical example might be this:

$$\sqrt{4x}$$

We can rewrite this expression according to the previous formula as follows:

$$\sqrt{4x}=\sqrt{4}\cdot\sqrt{x}=$$

And now we can square root the four:

$$=2\cdot\sqrt{x}$$

Similarly, we can rewrite an ordinary number if we want to keep the exact form. So an example:

$$\sqrt{18}=\sqrt{9}\cdot\sqrt{2}=3\sqrt{2}$$

Calculator

If you need to calculate a square root, you can use the square root calculator here 🧮.