Adding and Subtracting: Radicals

Adding and subtracting radicals involves mathematical expressions that contain a radical sign, indicating the root of a number. A radical is generally expressed in the form \( \sqrt[n]{x} \), where \( n \) represents the index and \( x \) is the radicand. Understanding the properties of radicals is essential, as they allow for simplifications such as the nth root of a product or quotient, and the manipulation of exponents.

For example, radicals can be simplified by recognizing that the square root of a product, \( \sqrt{xy} \), is equal to \( \sqrt{x} \times \sqrt{y} \). Square roots, typically represented without an index, always have a principal (positive) root, while non-negative integers that can be expressed as perfect squares hold specific properties as well.

When adding or subtracting radical expressions, it's necessary to simplify each term individually to reveal like terms that can be combined. This process entails ensuring that the radicands are the same, which allows for a more straightforward final expression. Understanding these fundamentals helps in performing operations with radicals effectively and accurately.

Published in: 2022

By:

Subject Terms

Adding and Subtracting: Radicals

A radical is a mathematical expression written in the form:

In this expression, n is called the index, and its value can be any integer greater than 1; x is called the radicand, and √ is called the radical sign. A radical can also be written in exponential form where:

Properties of Radicals

The nth root of xy is equal to the nth root of x multiplied by the nth root of y.

The nth root of x/y is equal to the nth root of x divided by the nth root of y.

The nth root of x^m is equal to the nth root of x raised to m.

The nth root of the mth root of x is equal to the (mn)th root of x.

Square Roots

For square roots, the index n, which is equal to 2,is not usually written next to the radical sign. Thus, square roots are usually expressed as , which is equal to the exponential expression .

A non-negative real numberx has two square roots, y and -y. However, the principal square root of x is the non-negative real number y. Thus, the principal square root is always the positive root.

A non-negative integer is a perfect square if it can be written in the form = x.

Simplifying Radicals

Below are some examples of how to simplify radicals.

Remember that the square root of xy is equal to the square root of x multiplied by the square root of y. Thus, the above expression is equal to

Simplifying the expression results in (9)(2), which is equal to 18. Therefore, = 18. Note that only the principal root (also known as the positive root) is written as the answer.

This can then be rewritten as

This is also equal to

Simplifying the expression results in . Therefore, = .

Note that for radicals that have indices n with odd values, there is only one root. In these cases, the sign (positive or negative) of the root is the same as the sign of the radicand.

Then, simplify the individual terms to get 5x² + 3x², which is equal to 8x². Therefore,

Note that at first, the terms did not seem like they could be combined because of the different indices (2 and 3). However, by simplifying the individual terms, the resulting terms ended up having the same variables. Thus, the final form of the expression is now only a single term.