Algebra 1 - Simplifying Radical Expressions


Recall the Basic Laws of Radicals:

1. ann=a or ann=a where a>0 if n is even.

If n is odd, then a and ann=a.

2. abn=an·bn where a>0 and b>0 if n is even

If n is odd, then a, b

3. abn=anbb where a>0 and b>0 if n is even.

If n is odd, then a, b but b0.

4. anm=amn

Rules in Simplifying Radicals:

  • Express the radicand in factored form. That is, write it as a product of the greatest perfect nth factor and another factor.
  • If the radical is a perfect square, factor the radicand, so the largest perfect square factor within the radicand is one of the factors. If the radical is a cube root, factor the radicand, so the largest perfect cube factor within the radicand is one of the factors, and so on.
  • Apply the multiplication law for radicals to show the expression as a product of two radical expressions.
  • Evaluate and simplify the radical expression completely. 

A simplified radical expression can be noticed if the following conditions are satisfied:

  • The radical expression has no perfect square factors or perfect nth factors aside from 1. A radical expression is left with irrational forms if the given radical expression is not a perfect nth root.
  • The index of the radical is expressed in its lowest possible form.
  • Fractions are no longer observed in the radicand. 
  • By rationalizing the denominator, radicals are no longer evident in the denominator.

Important Key Points

For any natural number n>1 and any real number a, we have: an=a1n.

Hence, an represents the nth root of a.

To change a+b23 in radical form, we have a+b23.

In a radical expression xn, we have the following conditions:

  • If x>0 and n is even, then we have a positive root (principal root) and a negative root.
  • If x>0 and n is odd, then we have a positive root. 
  • If x<0 and n is even, then there are no real roots.
  • If x<0 and n is odd, there is one negative root.
  • If x<0 and n is odd or even, then the root is equal to 0.

Steps in Solving Radical Equations:

  • Separate the radical expression on one side of the equation. Then, for cases of having a radical expression on both sides, raise both sides of the equation to the same positive integral power. 
  • After removing the radical symbols, solve the equation. Repeat the procedures if there are still radicals until you obtain an equation without radicals.
  • Solve the resulting equation and verify the obtained roots in the given equation.

Solved Examples

Example 1. What is the simplified form of 243x8y3z4?


81·3x8y2·y·z4=9x4yz2 3y


Example 2. What is the simplified form of 135x6y4z53?


27×5x6×y3×y×z3×z23=3x2yz 5yz23


Example 3. Simplify the radical expression 48a8b62a3b4.


48a8b62a3b4=24a5b2=4×6a4×a×b2=2a2b 6a


Example 4. Simplify the radical x183.




Example 5. FInd the solution(s) of the equation 4x-5=x+10.






Rationalizing the Denominator

Steps in Rationalizing the Denominator:

  • If the denominator is a monomial, multiply the numerator and denominator by an expression with factors that will make the exponents of the factors in the radicand of the denominator exactly divisible by the index.
  • If the denominator is a binomial, multiply the numerator and denominator by the conjugate of the denominator. 


Example 1. What is the simplified form of 149x3?




Example 2. Simplify the radical 2x+y.


The conjugate of x+y is x-y.


Cheat Sheet

  • In the expression 2x where x is a positive integer, the value of x must be any number divisible by the index 2 to generate an expression without radical. However, if all values of x that are not divisible by two are taken, these form an expression with radical.
  • The greatest perfect square factor of 1372x9 is 196x8, and the other factor is 7x.
  • The largest perfect cube factor of 1080ab4 is 216b3, and the other factor is 5ab.
  • A finite nested radical in the form a±2b can be simplified to a±b using the formula a±2b=a+b+2ab where a>b.
  • For infinite nested radicals a+a+a+a+..., we may use the shortcut method applying the formula x=121±4n+1 where n is the radicand and the sign ± depends on the operation indicated in the radicand.

Blunder Areas

  • The radical expression 76 is not yet in simplified form because a simplified radical must not contain a fractional radicand.
  • The radical expression 67 is not yet in simplified form since it has a radical in the denominator.
  • Not all obtained roots when solving radical equations satisfy the original equations. These roots/solutions are called extraneous roots.
  • Laws of integral exponents are also applicable in expressions involving fractional exponents.