Up

Adding and subtracting

Adding and subtracting can be a very easy process especially if you think back to when you first learned to add:

Example:

“If you have 2 apples and I give you 3 more, how many apples do you have?”

 

But is can get difficult sometimes unless you know what you’re doing and how to make it easier.

The addition/subtraction of radical expressions can be done just like “regular” numbers; however, in some cases you may not be able to simplify all the way down to one number.

 

To add or subtract radicals the must be like radicals.  Like radicals have the same root and radicand.

Radical-Expressions

Examples of like radicals are: \((\sqrt{2}, 5\sqrt{2}, -4\sqrt{2}) \) or \( ( \sqrt[3]{15}, 2\sqrt[3]{15}, -9\sqrt[3]{15}) \)

Simplify: 

\(3\sqrt{2} + 2\sqrt{2}\)

 

The terms in this expression contain like radicals so can therefore be added.  Now if you look at the radical part in the expression like the apple in the opening example about adding, you can get a better understanding of how to treat the radicals.  In other words:

3 “apples” + 2 “apples” = (3+2) “apples” or 5 “apples” (Now switch “apples” to \(\sqrt{2}\)).

\(3\sqrt{2} + 2\sqrt{2} = (3 + 2)\sqrt{2} = 5\sqrt{2}\)

 

Simplify: 

\(\sqrt{3} + 5\sqrt{3}\)

 

For the purpose of this explanation I will put the understood 1 in front of the first term giving me:

\(1\sqrt{3} + 5\sqrt{3}\)

(like radical terms can be combined)

\((1 + 5)\sqrt{3}\)

\(6\sqrt{3}\)

Simplify: 

\(-8\sqrt{5} + 5\sqrt{5}\) (like radical terms)

\((-8 + 5)\sqrt{5}\)

\(-3\sqrt{5}\)

Now what happens if we have unlike radicals?

Simplify: \(\sqrt{16} + \sqrt{4}\) (unlike radicals, so you can’t combine them…..yet)

Don’t assume that just because you have unlike radicals that you won’t be able to simplify the expression.  Therefore, in every simplifying radical problem, check to see if the given radical itself, can be simplified.

In this problem, the radicals simplify completely: \(\sqrt{16} + \sqrt{4} = 4 + 2 = 6\)

Simplify: 

\(2\sqrt{25} + 5\sqrt{9}\)

Think here that you have 2 square roots of 25 and 5 square roots of 9.

Now,\(\sqrt{25} = 5\) and \(\sqrt{9} = 3\) so,

\(2\sqrt{25} + 5\sqrt{9} = 2\cdot 5 + 5\cdot 3 = 10 + 15 = 25\)

Sometimes simplification isn’t as apparent.

Simplify: 

\(-2\sqrt{2} + \sqrt{18}\)

\(\sqrt{18}\) can be simplified (as seen in an earlier lesson): \(\sqrt{9\cdot 2} = \sqrt{9}\cdot \sqrt{2} = 3\sqrt{2}\)

Putting that back into the problem above yields:

\(-2\sqrt{2} + 3\sqrt{2} = -1\sqrt{2} = \sqrt{2}\)

 

Simplify: 

\(3\sqrt{7} - 2\sqrt{28} + 4\sqrt{7}\) (start by ensuring all radicals are simplified)

\(3\sqrt{7} - 2\sqrt{4\cdot 7} + 4\sqrt{7}\)

\(3\sqrt{7} - 2\cdot 2\sqrt{7} + 4\sqrt{7}\)

\(3\sqrt{7} - 4\sqrt{7} + 4\sqrt{7}\)

\((3 - 4 + 4)\sqrt{7} = 3\sqrt{7}\)

So you’re doing a problem and you’ve simplified your radicals; however, they’re not all alike.  WHAT DO YOU DO NOW?!?!?  Well, nothing………sort of.

Simplify: 

\(3\sqrt{3} + 2\sqrt{7} - \sqrt{3}\) (you notice that two of the three are alike, so combine the two like radical terms)

\(3\sqrt{3} - 1\sqrt{3} + 2\sqrt{7}\)

\((3 - 1)\sqrt{3} + 2\sqrt{7}\)

\(2\sqrt{3} + 2\sqrt{7}\) and that’s the answer!  It cannot be simplified any further.

Bellow you can download some free math worksheets and practice.


Downloads:
1395 x

Simplify.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Radical-Expressions-Adding-and-subtracting-easy

Watch below how to solve this example:

 

Downloads:
1289 x

Simplify.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Radical-Expressions-Adding-and-subtracting-mediumRadical-Expressions-Adding-and-subtracting-medium

Watch below how to solve this example:

 

Downloads:
1410 x

Simplify.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Radical-Expressions-Adding-and-subtracting-hard

Watch below how to solve this example:

 
 
 

Facebook PageGoogle PlusTwitterYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems