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Mathematics LibreTexts

6.2: Add and Subtract Radical Expressions

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Rules for radical multiplication and division have a simplicity and ease which lulls students into thinking addition and subtraction will follow suit. However, rules for addition and subtraction have more complication and less flexibility. Let’s start with the most common pitfall:

na+nbna+b

Note: The symbol says “does not equal.”

These two examples have even more temptation to go wrong due to the squaring and the variables:

x2+y2x+y

Also,

x2y2xy

This doesn’t change any property that was previously introduced! The following still holds true! (x0 and y0 if n is even)

x2+y2=x+y

And

x2y2=xy

And in general,

nxn+nyn=x±y

Use a calculator to convince yourself:

6+28

On the left side, find the value 6+2 to the nearest hundredth:

6+23.86

On the right side,

82.83

Now we see radicands cannot be added:

3.862.83

Use the order of operations as the problem is presented. Any other order of computation could be a pitfall.

327+38=3+2=5

The order of operations tells us to take the cube root first, then add. Use the order shown and it will be correct!

Using “Like Terms” to Add or Subtract “Like Radicands”

Adding and subtracting radicals starts with simplifying radicals. Use as often as possible the property nan=a to simplify radicals. We will continue to assume the variables take on nonnegative values only. Let’s explore how gathering like terms can be used with radicals.

2x+x=3x x= any real number

Gathering like terms is a technique you’ve used countless times before. Why not use with radicals? Radicals are real numbers, too! This approach opens our toolbox for adding or subtracting radical expressions. We can add radicals if the radicand is identical and index n also matches.

clipboard_e5b839b7d59c7487ff6af0c292561f2da.png

Suppose x=5. Gather like radicals:

25+5=35

Definition: Like Radicals

Two radical terms are said to be like radicals if they have the same index and the same radicand.

Example 6.2.1

Which of the three expressions can be simplified? Explain.

  1. 4310+7310
  2. 32+311
  3. 36+6

Solution

  1. This is the only given expression of the three that can be simplified.

4310+7310=11310

4310 and 7310 both contain 310 as the like radicals.

  1. Cannot be simplified.

32311

The radicands are not equal.

  1. Cannot be simplified.

366

The indices are not equal.

Simplify First

Don’t see a match? Make sure your radicals are completely simplified before you attempt to add or subtract radicals. The following examples demonstrate how simplification can help you find like radicals.

Example 6.2.2

Simplify 75+2123

Solution

75+21253At first glance, none of the radicals are like radicals. However, both 75 and 12 can be simplified.=253+2433Factor perfect squares within each radicand.=53+2233Simplify.=53+433Combine like radicals.(5+41)3=83

Example 6.2.3

Add 354x2+33x2+316x2

Solution

354x2+33x2+316x2Simplify each radical.=3272x2+33x2+382x2Factor the radicands.=332x2+33x2+233x2Two terms contain like radicals. Simplify. =533x2+33x2

Example 6.2.4

Find the area and the perimeter of the rectangle shown.

clipboard_e9e5cd0bd0869ddef6196db5a9138cd68.png

Length =(63+5) feet

Width =(3+55) feet

Solution

Use FOIL:

Area =(63+5)(3+55)=(63)(3)F+(63)(55)O+(5)(3)I+(5)(55)L=63+6535+53+55=18+3015+15+25=43+3115 ft2

Perimeter =P=2L+2W2(63+5)+2(3+55)=(263+25)Distributive Property+(23+255)Distributive Property=(123+25)+(23+255)=123+23Like Radicals+25+105Like Radicals=143+125 ft2

Try It! (Exercises)

For #1-4, add or subtract the expressions, if possible. Assume the variables represent nonnegative numbers.

  1. 7525+5
  2. 33x43x+66x86x
  3. 39y+49y32y+249y
  4. 7b516b24b316b23b53b2

For #5-8, simplify each radical expression, then add or subtract the expressions, if possible. Assume the variables represent nonnegative numbers.

  1. 23a3+5a3a27a3
  2. 356c62c23189+37c2
  3. 344p73p4324p3+2464p7
  4. 2b5192b3564b+356b8+5486b

For #9-14, evaluate each of the following. Simplify wherever possible. Assume the variables represent nonnegative numbers.

  1. 12(218)
  2. (7310+334)(2320350)
  3. (92x+63x)2
  4. (26410)(26+410)
  5. (324354)2
  6. 5(371)(414+1)

For #15-16, find the area and the perimeter of each rectangle. The length and width are each marked in the figures and the units are feet.

  1. clipboard_ee3bb371de39b456d6c73cfeeb3def10f.png
  2. clipboard_e2ae9bd4a09dde96bb8e3712b6624465a.png
  3. A square has side lengths = (44+2) feet. Find each of the area and the perimeter of the square.
  4. If a square has area =540 ft2. What is the perimeter of the square?

This page titled 6.2: Add and Subtract Radical Expressions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.

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