# Radical Expression: Definition, Examples & Quiz

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*A radical expression is any mathematical expression containing a radical symbol (√). This lesson will go into more detail about the types of radical expressions and give some examples on how to work with them in mathematics. There will be a quiz at the end of the lesson.*

## Definition of Radical Expression

In mathematics, a radical expression is defined as any expression containing a **radical** (√) symbol. Many people mistakenly call this a 'square root' symbol, and many times it is used to determine the square root of a number. However, it can also be used to describe a cube root, a fourth root or higher. When the radical symbol is used to denote any root other than a square root, there will be a superscript number in the 'V'-shaped part of the symbol. For example, 3√(8) means to find the **cube root** of 8. If there is no superscript number, the radical expression is calling for the square root.

The term underneath the radical symbol is called the **radicand**.

## History of the Term Radical

The terms 'radical' and 'radicand' are both derived from the Latin word 'radix,' which means 'root'. The reason for this is that the root is the source of something (like the root of a word); if you square or cube a number, the number that it came from is the root, while the number itself (the radicand) grows from that root. The first usage of these terms was seen in England in the mid 1600s. They were first used in a book called *An Introduction to Algebra* by John Pell.

## Problem Solving

To solve a problem involving a square root, simply take the square root of the radicand. The square root of a number is the number that, when multiplied by itself, or **squared** is equal to the radicand.

For example, √(25) = 5 because 5 x 5 = 25

If there is a subscript number in front of the radical symbol, that number tells you how many times a number should be multiplied by itself to equal the radicand. It is the opposite of an exponent, just like addition is the opposite of subtraction or division is the opposite of multiplication.

Back to the example in the first paragraph with the cube root of 8, 3√(8) = 2 because 2^3 = 8 or 2 x 2 x 2 = 8.

Similarly, 5√(243) = 3 because 3^5 = 243 (3 x 3 x 3 x 3 x 3 = 243).

## Examples

1.) Solve √(64)

Since 8^2 = 64 the square root of 64 is 8, so the answer to this problem is 8.

2.) Solve 3√(343)

The cube root of 343 is 7 (7^3 = 343)

3.) 5√(243)

3 (3^5 = 243)

## Special Cases

There are certain circumstances where finding the root of a number is impossible or the result might be something unexpected.

One case is with negative numbers. If the root needed is an even number, such as the square root or fourth root, the root is an imaginary number. There is no real root. This is because there is no number that can be multiplied to itself to equal a negative number. If the problem is looking for an odd root, the radicand can be negative. This is because a number multiplied three times (or five, or any other odd amount) can be negative. For example, -3 * -3 * -3 = -27.

If the radicand is 1, then the answer will be 1, no matter what the root is. This is because 1 times itself is always 1.

A radicand of 0 results in an answer of 0, no matter the root.

## Conclusion

Radical expressions are mathematical expressions that contain a √. It does not have to be a square root, but can also include cube roots, fourth roots, fifth roots, etc. The way to determine which type of root is by the superscript number in front of the radical. Finding the root of a number is the opposite operation from raising a number to a power. To find the root of a number, just determine which number that when raised to the power defined by the superscript is equal to the radicand.

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