# How to Simplify Radical Expressions

Six Methods:Perfect PowersConvert Rational Exponents to RadicalsRemove Fractions from RadicalsCombine Products of RadicalsExtract Square Factors from RadicalsRationalize the Denominator

A radical expression is an algebraic expression that includes a square root (or cube or higher order roots). Often such expressions can describe the same number even if they appear very different (ie, 1/(sqrt(2) - 1) = sqrt(2)+1). The remedy is to define a preferred "canonical form" for such expressions. If two expressions, both in canonical form, still look different, then they indeed are unequal. Mathematicians agreed that the canonical form for radical expressions should:

- Avoid fractions in radicals
- Not use fractional exponents
- Avoid radicals in denominators
- Not multiply radicals by radicals
- Have only squarefree terms under the radicals

One practical use for this is in multiple choice exams. When you've solved a problem, but your answer doesn't match any of the multiple choices, try simplifying it into canonical form. Since test writers usually put their answers in canonical form, doing the same to yours will make it apparent which of their answers is equal to yours. In free-response exams, instructions like "simplify your answer" or "simplify all radicals" mean the student is to apply these steps until their answer satisfies the canonical form above. It is also of some use in equation solving, although some equations are easier to deal with using a non-canonical form.

## Steps

- 1
**If necessary, review the rules for manipulation of radicals and exponents (they're the same - roots are fractional powers) as most of them are needed for this process.**Also review the rules for manipulating and simplifying polynomial and rational type expressions as they too will be needed throughout to simplify.

### Method 1 Perfect Powers

- 1
**Simplify any radical expressions that are perfect squares.**A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. To simplify a perfect square under a radical, simply remove the radical sign and write the number that is the square root of the perfect square.- For example, 121 is a perfect square because 11 x 11 is 121. Thus, you can simplify sqrt(121) to 11, removing the square root symbol.
- To make this process easier, you should memorize the first twelve perfect squares: 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, 6 x 6 = 36, 7 x 7 = 49, 8 x 8 = 64, 9 x 9 = 81, 10 x 10 = 100, 11 x 11 = 121, 12 x 12 = 144

- 2
**Simplify any radical expressions that are perfect cubes.**A perfect cube is the product of any number that is multiplied by itself twice, such as 27, which is the product of 3 x 3 x 3. To simplify a radical expression when a perfect cube is under the cube root sign, simply remove the radical sign and write the number that is the cube root of the perfect cube.- For example, 343 is a perfect cube because it is the product of 7 x 7 x 7. Therefore, the cube root of the perfect cube 343 is simply 7.

### Method 2 Convert Rational Exponents to Radicals

Or convert the other way if you prefer (sometimes there are good reasons for doing that), but don't mix terms like sqrt(5) + 5^(3/2) in the same expression. We will assume that you decide to use radical notation and will use sqrt(n) for the square toot of n and cbrt(n) for cube roots.

- 1
**Find any fractional exponent and convert it to the radical equivalent, namely x^(a/b) = bth root of x^a**- If you have a fraction for the index of a radical, get rid of that too. For instance the (2/3) root of 4 = sqrt(4)^3 = 2^3 = 8.

- 2
**Convert negative exponents to their equivalent fraction, namely x^-y = 1/x^y**- This only applies to constant, rational exponents. If you have terms like 2^x, leave them alone, even if the problem context implies that x might be fractional or negative.

- 3
**Combine any like terms and simplify any rational expressions that result.**

### Method 3 Remove Fractions from Radicals

Canonical form requires expressing the root of a fraction in terms of roots of whole numbers

- 1
**Examine terms under each radical to see if any contain fractions.**If so, ... - 2
**Replace it as a ratio of two radicals using the identity sqrt(a/b) = sqrt(a)/sqrt(b).**- Don't use this identity if the denominator is negative, or is a variable expression that might be negative. In that case, simplify the fraction first.

- 3
**Simplify any perfect squares that result.**That is, convert sqrt(5/4) to sqrt(5)/sqrt(4), and then further simplify it to sqrt(5)/2. - 4
**Make any other useful simplifications such as reducing compound fractions, combining like terms, etc.**

### Method 4 Combine Products of Radicals

- 1
**If you have one radical expression multiplied by another, combine them as a single radical using the property:**sqrt(a)*sqrt(b) = sqrt(ab). For example, replace sqrt(2)*sqrt(6) by sqrt(12).- The above identity, sqrt(a)*sqrt(b) = sqrt(ab) is valid for non negative radicands. Don't apply it if a and b are negative as then you would falsely assert that sqrt(-1)*sqrt(-1) = sqrt(1). The left-hand side -1 by definition (or undefined if you refuse to acknowledge complex numbers) while the right side is +1. If a and/or b is negative, first "fix" its sign by sqrt(-5) = i*sqrt(5). If the radicand is a variable expression whose sign is not known from context and could be either positive or negative, then just leave it alone for now. You could use the more general identity, sqrt(a)*sqrt(b) = sqrt(sgn(a))*sqrt(sgn(b))*sqrt(|ab|) which is valid for all real numbers a and b, but it's usually not worth the added complexity of introducing the sign function.
- This identity only applies if the radicals have the same index. You can multiply more general radicals like sqrt(5)*cbrt(7) by first expressing them with a common index. To do this, temporarily convert the roots to fractional exponents: sqrt(5)*cbrt(7) = 5^(1/2) * 7^(1/3) = 5^(3/6) * 7^(2/6) = 125^(1/6) * 49^(1/6). Then apply the product rule to equate this product to the sixth root of 6125.

### Method 5 Extract Square Factors from Radicals

- 1
**Factorize an imperfect radical expression into its prime factors.**The factors are the numbers that multiply to create a number -- for example, 5 and 4 are two factors of the number 20. To break down an imperfect radical expression, write down all of the factors of that number (or as many as you can think of, if it's a large number) until you find one that is a perfect square.- For example, try listing all the factors of the number 45: 1, 3, 5, 9, 15, and 45. 9 is a factor of 45 that is also a perfect square (9=3^2). 9 x 5 = 45.

- 2
**Remove any multiples that are a perfect square out of the radical sign.**9 is a perfect square because it is the product of 3 x 3. Take the 9 out of the radical sign and place a 3 in front of it, leaving 5 under the radical sign. If you "throw" the three back in under the radical sign, it will be multiplied by itself to create 9 again, which will multiply with 5 to create 45 again. 3 root 5 is just a simplified way of saying root 45.- That is, sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5).

- 3
**Find a perfect square in the variable.**The square root of*a*to the second power would be*|a|.*You can further simplify this to just "a" only if the variable is known to be positive. The square root of*a*to the third power is broken down into the square root of*a*squared times*a*-- this is because you add exponents when you multiply variables, so that*a*squared times*a*is equal to*a*cubed.- Therefore, the perfect square in the expression
*a*cubed is*a*squared.

- Therefore, the perfect square in the expression
- 4
**Pull any variables that are perfect squares out of the radical sign.**Now, take*a*squared and pull it out of the radical to make it a regular*|a|*. The simplified form of*a*cubed is just*|a|*root*a.* - 5
**Combine any like terms and simplify any rational expressions that result.**

### Method 6 Rationalize the Denominator

Canonical form requires the denominator to be a whole number (or a polynomial if it contains indeterminate) if at all possible.

- 1
- If the denominator consists of a single term under a radical, such as [stuff]/sqrt(5), then multiply numerator and denominator by that radical to get [stuff]*sqrt(5)/sqrt(5)*sqrt(5) = [stuff]*sqrt(5)/5.
- For cube or higher roots, multiply by the appropriate power of the radical to make the denominator rational. If the denominator was cbrt(5), then multiply numerator and denominator by cbrt(5)^2.

- If the denominator consists of a sum or difference of square roots such as sqrt(2) + sqrt(6), then multiply numerator and denominator by its conjugate, the same expression with the opposite operator. Thus [stuff]/(sqrt(2) + sqrt(6)) = [stuff](sqrt(2)-sqrt(6))/(sqrt(2) + sqrt(6))(sqrt(2)-sqrt(6)). Then use the difference of squares identity [(a+b)(a-b) = a^2-b^2] to rationalize the denominator, simplifying (sqrt(2) + sqrt(6))(sqrt(2)-sqrt(6)) = sqrt(2)^2 - sqrt(6)^2 = 2-6 = -4.
- This works for denominators like 5 + sqrt(3) too since every whole number is a square root of some other whole number. [1/(5 + sqrt(3)) = (5-sqrt(3))/(5 + sqrt(3))(5-sqrt(3)) = (5-sqrt(3))/(5^2-sqrt(3)^2) = (5-sqrt(3))/(25-3) = (5-sqrt(3))/22]
- This works for a sum of square roots like sqrt(5)-sqrt(6)+sqrt(7). If you group it as (sqrt(5)-sqrt(6))+sqrt(7) and multiply it by (sqrt(5)-sqrt(6))-sqrt(7), your answer won't be rational, but will be of the form a+b*sqrt(30) where a and b are rational. Then you can repeat the process with the conjugate of a+b*sqrt(30) and (a+b*sqrt(30))(a-b*sqrt(30)) is rational. In essence, if you can use this trick once to reduce the number of radical signs in the denominator, then you can use this trick repeatedly to eliminate all of them.
- This even works for denominators containing higher roots like the 4th root of 3 plus the 7th root of 9. Just multiply numerator and denominator by the denominator's conjugate. Unfortunately, it is not immediately clear what the conjugate of that denominator is nor how to go about finding it. A good book on algebraic number theory will cover this, but I will not.

- If the denominator consists of a single term under a radical, such as [stuff]/sqrt(5), then multiply numerator and denominator by that radical to get [stuff]*sqrt(5)/sqrt(5)*sqrt(5) = [stuff]*sqrt(5)/5.
- 2
**Now the denominator is rationalized, but the numerator is a mess.**You now have whatever you started with up there times the denominator's conjugate. Go ahead and expand that product like you would for a product of polynomials. See if anything cancels or simplifies and combine like terms if possible. - 3
**If the denominator is a negative integer, then multiply numerator and denominator by -1 to make it positive.**

## Tips

- There are websites that you can search online that will simplify a radical expression for you. You simply type in the equation under the radical sign, and after hitting enter, your simplified answer will appear.
- For simple problems, many of these steps won't apply. For complicated problems, some of them may need to be applied more than once. Make "easy" simplifications continuously as you work, and check your final answer against the canonical form criteria in the intro. If your answer is canonical, you are done; while it is not canonical, one of these steps will tell you what still needs to be done to make it so.
- Most references to the "preferred canonical form" for a radical expression also apply to complex numbers (i = sqrt(-1)). Even if it's written as "i" rather than with a radical sign, we try to avoid writing i in a denominator.
- Parts of these instructions assume that all radicals are square roots. The general principles are the same for cube or higher roots, although some of them (particularly rationalizing the denominator) may be harder to apply. You'll also have to decide if you want terms like cbrt(4) or cbrt(2)^2 (I can't remember which way the textbook authors prefer).
- Parts of these instructions misuse the term "canonical form" when they actually describe only a "normal form". The difference is that a canonical form would require either 1+sqrt(2) or sqrt(2)+1 and label the other as improper; a normal form assumes that you, dear reader, are bright enough to recognize these as "obviously equal" as numbers even if they aren't typographically identical (where 'obvious' means using only arithmetical properties (addition is commutative), not algebraic properties (sqrt(2) is a non-negative root of x^2-2)). We hope readers will forgive this mild abuse of terminology.
- If these instructions seem ambiguous or contradictory, then apply all consistent and unambiguous steps and then choose whatever form looks most like the way radical expressions are used in your text.

## Article Info

Categories: Algebra