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How to Add Radicals

Narayani Karthik May 13, 2019
A radical is a mathematical term which means 'root'. Square roots and cube roots can be added together. This post will deal with adding square roots.
When we look at mathematical equations like 3x3=9 or 3x3x3=27, what does it imply? Both equations have one thing in common, and that is number 3. Rephrase the equations and it becomes: 32=9 and 33=27. One thing that is clear from both these equations is that a changing exponent value yields results of 9 and 27.
Exponents and radicals are important connotations in math, precisely algebra. Exponents are a convenient way of expressing repetitive multiplications. Whereas, radicals (often known as surds) are symbols that represent the exponents.
So for the given example, the exponent expression using radicals will be represented as: √27, which can be further simplified as √(3x3x3) = 3√3, where 3 is the root number, which is covered by the radical symbol.

Basic Rules

Before we get to learn about adding radicals, one must know some rules. A radical to be computed, must be in its simplified form, with no more square roots, cube roots or fourth roots. Here are some rules to be kept in mind:
  • n√a = a1/n
  • (n√a)m = (a1/n)m = (am/n)
  • n√an = a (n-th root of a Number to the Power n)
  • n√a x n√b =n√ab (The product of the n-th root of a and the n-th root of b is the n-th root of ab)
  • m(√n√a) = mn√a (The m-th Root of the n-th Root of the Number a is the mn-th Root of a)
  • n√a / n√b = n√(a/b) (The n-th Root of an Over the n-th Root of b is the n-th Root of a/b)
Here are some standard rules to be followed when adding radicals, which are considered similar:
  • Both radicals should have the same index.
  • The quantities under the radical sign √ should be same.
  • Variables outside the radicals are also same.
Follow the fundamental algebra tips, and your concepts about radicals will be clear.

Adding Radicals with Variables

Here is an example of adding radicals with variables, which will help you apply the given rules to solve an equation.

Example 1:
√5 + 2√3 - 5√5
=>2√3 - 5√5 + √5 (simplifying the same radicands)
=>2√3 - 4√5

Example 2:
3b√(27a5b) + 2a√(3a3b3)
In this example, if you take a close look, the index is same, but the quantities of the radical are not. Also, the radicals outside are not the same. So, you need to simplify them here, by resolving off all the square and cube roots.
This is how you can simplify the equation:

3b√(27a5b) + 2a√(3a3b3)
=> 3b√(27a.a.a.a.a.b) + 2a√(3a.a.a.b.b.b) (Here (.) implies multiplication)
=> 3b√(3.(3.3) (a.a) (a.a) a.b)) + 2a√(3 a(a.a) b(b.b))
=> 3b.3.a.a√3ab + 2a.a.b√3ab
=> 9a2b√3ab + 2a2b√3ab

Adding Radicals with Fractions

For fractions too the rules are the same. Simplification precedes adding. Have a look at this example to get a better understanding.
Example 1:
3/√5 + √5
The first step is to rationalize and bring the same denominators, while adding fractions.
3/√5 + √5
=> (3 + (√5.√5))/√5
=> (3 + 5)/√5
=> 8/√5 (make the denominator a complete rational number)
=> (8/√5) (√5/√5) = 8√5/5
Example 2:
√(a3/b5) + √ba

This is an algebraic expression which first has to be simplified.
√(a3/b5) + √ba
=> √((a.a.a)/(b.b.b.b.b)) + √ba
=> √((a.a.a.b)/(b.b.b.b.b.b)) + √ba (multiplying b in both numerator and denominator of the first radical)
=> a/b3(√ab) + √ba
=> √ab (a/b3 + 1)
=> √ab (a + b3)/b3
Adding exponents and radicals are quite useful in simplifying complex algebraic expressions. They are also used in complicated equations in astronomy and physics. Now, solving complex algebraic expressions will be a cakewalk for you.