Adding Radical Expressions With Variables

Adding and Subtracting Radicals

Learning Objective(s)

· Add radicals and simplify.

· Subtract radicals and simplify.

Introduction

There are 2 keys to combining radicals past addition or subtraction: expect at the index, and look at the radicand. If these are the aforementioned, then addition and subtraction are possible. If not, then you cannot combine the two radicals.

Making sense of a string of radicals may be difficult. 1 helpful tip is to call up of radicals as variables, and treat them the same fashion. Let's showtime at that place.

Thinking about Radicals equally Variables

Radicals can look confusing when presented in a long cord, as in . How do yous simplify this expression? (It is worth noting that you will not often see radicals presented this fashion…but it is a helpful fashion to innovate calculation and subtracting radicals!)

Treating radicals the same way that y'all treat variables is oftentimes a helpful identify to outset. For case, you would accept no problem simplifying the expression below.

Combining similar terms, y'all can quickly notice that three + ii = v and a + 6a = viia. The expression can exist simplified to 5 + 7a + b.

The same is truthful of radicals. Equally long every bit radicals have the same radicand (expression under the radical sign) and index (root), they can be combined. Beneath, the two expressions are evaluated side by side.

Here'due south another style to think about it. Recall that radicals are but an alternative way of writing fractional exponents. So, for example, , and . If you call up of radicals in terms of exponents, then all the regular rules of exponents utilize.

Expect at the expressions beneath. On the left, the expression is written in terms of radicals. On the correct, the expression is written in terms of exponents.

So what does all this mean? Well, the bottom line is that if you need to combine radicals past adding or subtracting, make sure they have the aforementioned radicand and root. And if things get disruptive, or if you merely desire to verify that yous are combining them correctly, you tin can ever use what you know about variables and the rules of exponents to help you lot.

Adding Radicals

Permit'south look at some examples. In this commencement example, both radicals have the same root and index.

Example
Trouble	Add.
		The two radicals are the same, . This means you tin combine them equally yous would combine the terms .
Respond

This next example contains more addends. Notice how yous can combine like terms (radicals that have the same root and alphabetize) but y'all cannot combine dissimilar terms.

Instance
Problem	Add together.
		Rearrange terms so that similar radicals are next to each other. And then add.
Answer

Notice that the expression in the previous example is simplified fifty-fifty though it has two terms: and . Information technology would be a fault to endeavour to combine them farther! (Some people make the mistake that . This is incorrect because and are non like radicals so they cannot exist added.)

Example
Problem	Add.
		Rearrange terms then that similar radicals are next to each other. And so add.
Answer

Sometimes you may demand to add together and simplify the radical. If the radicals are different, try simplifying first—you may end upward being able to combine the radicals at the stop, as shown in these next 2 examples.

Example
Trouble	Add together and simplify.
		Simplify each radical by identifying perfect cubes.


		Simplify.
		Add together.
Answer

Case
Problem	Add and simplify.
		Simplify each radical by identifying perfect cubes.


		Add like radicals.
Answer

Add.

Show/Hide Answer

Correct. When adding radical expressions, you tin can combine like radicals just as yous would add like variables. .

Incorrect. You reversed the coefficients and the radicals. The correct answer is .

Incorrect. Remember that you cannot add radicals that have different index numbers or radicands. Identify like radicals in the expression and try adding again. The right answer is .

Incorrect. Recollect that you cannot add 2 radicals that have unlike alphabetize numbers or radicands. Identify like radicals in the expression and try calculation over again. The right answer is .

Subtracting Radicals

Subtraction of radicals follows the aforementioned set of rules and approaches as addition—the radicands and the indices (plural of index) must be the same for two (or more) radicals to be subtracted. In the 3 examples that follow, subtraction has been rewritten as addition of the opposite.

Example
Problem	Subtract.
		The radicands and indices are the aforementioned, and then these 2 radicals tin be combined.
Answer

Example
Trouble	Subtract.
		Two of the radicals have the aforementioned alphabetize and radicand, then they tin can be combined. Rewrite the expression then that similar radicals are next to each other.
		Combine. Although the indices of and are the aforementioned, the radicands are non—so they cannot be combined.
Answer

Case
Problem	Subtract and simplify.
		Simplify each radical by identifying and pulling out powers of four.


Answer

Subtract and simplify.

Prove/Hide Answer

Incorrect. To simplify, y'all tin rewrite as . Then pull out the foursquare roots to get The correct answer is .

Wrong. Recall that you cannot combine two radicands unless they are the same. , only . The correct answer is .

Right. Rewriting every bit , you lot constitute that .

Wrong. To simplify, you tin rewrite as . Then pull out the square roots to get . The right reply is .

Summary

Combining radicals is possible when the alphabetize and the radicand of two or more radicals are the aforementioned. Radicals with the same alphabetize and radicand are known equally like radicals. Information technology is often helpful to treat radicals just as you would treat variables: like radicals can be added and subtracted in the same way that like variables can be added and subtracted. Sometimes, you will need to simplify a radical expression before information technology is possible to add or decrease like terms.