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ACT Prep: Help and Review42 chapters | 399 lessons

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Instructor:
*Jennifer Beddoe*

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.

In mathematics, a **radical expression** is defined as any expression containing a **radical** (âˆš) 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, 3âˆš(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**.

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.

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, âˆš(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 with the cube root of 8, 3âˆš(8) = 2 because 2^3 = 8 or 2 x 2 x 2 = 8.

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

1.) Solve âˆš(64)

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

2.) Solve 3âˆš(343)

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

3.) 5âˆš(243)

3 (3^5 = 243)

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.

**Radical expressions** are mathematical expressions that contain a âˆš. 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, when raised to the power defined by the superscript, is equal to the **radicand**.

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ACT Prep: Help and Review42 chapters | 399 lessons

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Radicands and Radical Expressions 4:29
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Simplify Square Roots of Quotients 4:49
- Rationalizing Denominators in Radical Expressions 7:01
- Addition and Subtraction Using Radical Notation 3:08
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Negative Square Root: Definition & Overview 8:13
- Radical Expression: Definition & Examples 4:00
- Go to ACT Math - Radicals: Help and Review

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