Addition and subtraction of radicals

The addition and subtraction of radicals is similar to the addition and subtraction of
algebraic expressions with like terms. 

Notice how the distributive property is used to combine 6x and 2x.

6x + 2x = (6 + 2)x = 8x

The distributive property can also be used to add and subtract expressions containing radicals.

$$ 3\sqrt{2} + 7 \sqrt{2} = (3 + 7)\sqrt{2} = 10 \sqrt{2} $$

$$ We \ call \ 3\sqrt{2}\ and \ 7 \sqrt{2} \ like\ radicals $$


Like radicals

Like radicals are radical expressions that have the same radicand and the same index.

Only like radicals can be added or subtracted using the distributive property. The following radical expressions cannot be added or subtracted.

$$ a. \ 3\sqrt{2} + 2 \sqrt{3} $$
$$ b. \ 8\sqrt[3]{3} + 5 \sqrt{3} $$

a. has the same index, but the radicand is not the same.

b. has the same radicand, but the index is not the same.


More examples of addition and subtraction of radicals

$$ 6\sqrt{5} + 3 \sqrt{5} = (6 + 3)\sqrt{5} = 9 \sqrt{5} $$


$$ 18\sqrt[3]{7} - 5 \sqrt[3]{7}- \sqrt[3]{7} = (18 - 5 - 1)\sqrt[3]{7} $$
$$ 18\sqrt[3]{7} - 5 \sqrt[3]{7} - \sqrt[3]{7} = 12\sqrt[3]{7} $$

Sometimes, you may need to simplify each radical until you get the same radicand before you add and / or subtract radicals. The next example demonstrates how.

$$ \sqrt{50} + \sqrt{8} = \sqrt{25 \times 2} + \sqrt{4 \times 2} $$
$$ \sqrt{50} + \sqrt{8} = \sqrt{25} \times \sqrt{ 2} + \sqrt{4} \times \sqrt{2} $$
$$ \sqrt{50} + \sqrt{8} = 5\sqrt{ 2} + 2\sqrt{2} $$
$$ \sqrt{50} + \sqrt{8} = (5 + 2) \sqrt{ 2} $$
$$ \sqrt{50} + \sqrt{8} = 7 \sqrt{ 2} $$

Homepage

Algebra lessons

Adding and subtracting radicals

Recent Articles

RSS

  1. Operations on Functions

    Apr 01, 20 01:18 PM

    Learn how to perform operations on functions such as adding, subtracting, multiplying, and dividing functions

    Read More

Check out some of our top basic mathematics lessons.

Formula for percentage

Finding the average

Basic math formulas

Algebra word problems

Types of angles

Area of irregular shapes

Math problem solver

Math skills assessment

Compatible numbers

Surface area of a cube

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

                                 Follow me on Pinterest

Follow me on Pinterest

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.