Integer

In order to understand radicals, you should first know what an integer is.  An integer is part of a group of numbers known as real numbers. There are two main types of numbers that are not real numbers:

  • Imaginary numbers
  • Infinity

 

Remember, these are not real numbers. You are probably more familiar with real numbers, which include:

  • Whole numbers
  • Rational numbers
  • Irrational numbers

 

Take a look at the image and table below to better understand the difference between these two types of numbers.

 

real_numbers_not_real

 

Description More examples
Whole Positive real number without decimal/fraction 188, 7 000
Rational Positive or negative with decimal/fraction 1.1, -888.99
Irrational Positive or negative, can’t be written as decimal/fraction \sqrt{3}, \sqrt[^3]{81}

 

You may be wondering, what happens when you have a negative whole number? That is actually an integer.

Real Number

Real numbers are the basis of most of the maths you will learn or use. Whenever you’re in doubt, it can be helpful to look at a number line, also called a “real number line.”

 

number_line_origin

 

As you can see, the real number line includes both positive and negative numbers.

Exponents

We’re almost to radicals. One last concept to understand before diving into radicals is that of an exponent. An exponent can be thought of as the opposite of a radical. Another name for exponents is ‘powers.’ The table below describes some exponents.

Meaning Common Name Written
2^{2} 2 x 2 Square Two to the power of 2; Two squared
3^{3} 3 x 3 x 3 Cube 3 to the power of 3; 3 cubed
4^{2} 4 x 4 Square 4 to the power of 2; 4 squared

Radical Definition

Recall that a radical, also called a ‘root,’ is the opposite of an exponent. Because of this, you can think of each operation as ‘undoing’ the other. Let’s first go through the most common radical: the square root.

Square Root Result Written
2^{2} = 4 \sqrt{4}, \sqrt{2 x 2} 2 Square root of 2
7^{2} = 49 \sqrt{49}, \sqrt{7 x 7} 7 Square root of 3
4^{2} = 16 \sqrt{16}, \sqrt{4 x 4} 4 Square root of 2

Notice what the operations in the table have in common? The square roots are all ‘undoing’ powers of 2. In other words, the square root takes any number and tries to find what number ‘fits’ exactly twice when multiplied by itself.

When you want to take the nth root of a number x, it means you want to find the number k that, multiplied by itself n times, will give you the original number x.

 

radical_roots

 

Take a look at some examples below. 

Square Root Result Result
\sqrt[^3]{27} \sqrt[^3]{27} = k * k * k \sqrt[^3]{3x3x3} 3
\sqrt[^4]{16} \sqrt[^4]{16} = k * k * k * k \sqrt[^4]{4x4x4x4} 4
\sqrt[^5]{243} \sqrt[^3]{2401} = k * k * k * k * k \sqrt[^5]{5x5x5x5x5} 5

Notation of Radicals

Radicals have some common notations. The first thing you should note is that a radical can actually be written as an exponent. Take a look below.

radical_rules

 

While this may seem confusing, we break down each below.

Explanation Example
\sqrt{x} = \sqrt[^2]{x} A square root is simply a number to the 2nd root, because they are the most common the 2 is usually not written \sqrt{4} = \sqrt[^2]{4}
\sqrt[^n]{x} A number to the nth root \sqrt[^n]{16}
x^{\frac{1}{n}} = \sqrt[^]{} A number to the nth root can also be written as an exponent, which is 1 over the nth root 27^{\frac{1}{3}} = \sqrt[^3]{27}

Radical Example

Let’s take a look at some examples of radicals. Let's complete the following task: convert the sentence into the common notation for the radical.

  • The 4th root of 16
  • The square root of 225
  • 4 to the power of \fra{1}{2}

After this, solve the problem. These kinds of problems are often solved by calculators, which usually have the square root symbol and a separate symbol: \sqrt[^n]{x}.

 

roots_rules

Radical Rules

There are a couple of rules you should keep in mind when you’re dealing with radicals. These rules are written in the table below, along with their descriptions.

Rule Example
\sqrt[^n]{x^{n}} = x \sqrt[^3]{15^{3}} = 15
\sqrt[^n]{xy} = \sqrt[^n]{x} * \sqrt[^n]{y} \sqrt[^4]{16*27} = \sqrt[^4]{16} * \sqrt[^4]{27} = 2*3 = 6
\sqrt[^n]{\frac{x}{y}} = \frac{\sqrt[^n]{x}}{\sqrt[^n]{y}} \sqrt[^2]{\frac{49}{16}} = \frac{\sqrt[^2]{49}}{\sqrt[^2]{16}} = \frac{7}{4}
\sqrt{x}*\sqrt{x} = x \sqrt{400}*\sqrt{400} = 400

Radical Simplification

There are many ways that you can simplify a radical. The most common way you can try to simplify a radical is to try and divide the number up inside of the radical so that you can get a perfect square.

A perfect square is a radical that contains a number that can be simplified into a square. Take four as an example.

 

square_root

 

As you can see, the number 4 can be broken down into 2 squared. We look for these types of numbers inside of a radical because they are easy to take out.

To simplify larger radicals, you can apply the same approach. Take the number 20 as an example.

 

square_multiple

 

As you can see, we first:

  • Break down the original number
  • See if there are any perfect squares
  • We are left with 2 * \sqrt{5}

Radical Division

In order to simplify radical division, we need to use a combination of the rules we learned above. First, we have the following.

    \[ \sqrt{\frac{4}{32}} \]

We use the rules above to solve by first rewriting it, then splitting it into smaller numbers.

    \[ \frac{\sqrt{4}}{\sqrt{16*2}} = \frac{\sqrt{2^2}}{\sqrt{4^2} * {\sqrt{2}}} = \frac{2}{4*{\sqrt{2}}} \]

Simplifying the nominator and denominator, we simply get:

    \[ \frac{1}{2*{\sqrt{2}}} \]

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.