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Simplifying Radical Expressions

Simplifying Radical Expressions

Author: Sophia Tutorial

This lesson relates properties of exponents to properties of radical expressions, and applies the properties to simplifying expressions involving radicals. 

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Simplifying Radical Expressions

Simplifying Radical Expressions

Square roots are the most common type of radical used. A square root “unsquares” a number. For example, because 52 = 25 we say the square root of 25 is 5. The square root of 25 is written as √25.

The radical sign, when first used was an R with a line through the tail, similar to our prescription symbol today. The R came from the latin, “radix”, which can be translated as “source” or “foundation”. It wasn’t until the 1500s that our current symbol was first used in Germany (but even then it was just a check mark with no bar over the numbers!).

Not all numbers have a nice even square root. For example, if we found √8 on our calculator, the answer would be 2.828427124746190097603377448419... and even this number is a rounded approximation of the square root. Decimal approximations are good estimates, but you may need the exact value.  To do this, we will simplify radical expressions using the product rule of radicals:

Product Property of Square Roots: square root of a b end root space equals space square root of a space • space space root index blank of b

We can use the product rule to simplify an expression such as √180 by splitting it into two roots, √36 . √5, and simplifying the first root, 6√5. The trick in this process is being able to translate a problem like √180 into √36 • √5. There are several ways this can be done.

The most common and, with a bit of practice, the fastest method, is to find perfect squares that divide evenly into the radicand, or number under the radical. This is shown in the next example.

If there is a coefficient in front of the radical to begin with, the problem merely becomes a big multiplication problem.

As we simplify radicals using this method it is important to be sure our final answer can be simplified no more.

The previous example could have been done in fewer steps if we had noticed that 72=36 times 2, but often the time it takes to discover the larger perfect square is more than it would take to simplify in several steps.

We can simplify higher roots in much the same way we simplified square roots, using the product property of radicals.

Product Property of Radicals: m-th root of a b end root space equals space space m-th root of a space • space space m-th root of b

Often we are not as familiar with higher powers as we are with squares. It is important to remember what index we are working with as we try and work our way to the solution.

Just as with square roots, if we have a coefficient, we multiply the new coefficients together.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at:

Formulas to Know
Product Property of Radicals

n-th root of a b end root space equals space n-th root of a space times space n-th root of b

Product Property of Square Roots

square root of a b end root equals square root of a times square root of b

Quotient Property of Radicals

n-th root of a over b end root equals fraction numerator n-th root of a over denominator n-th root of b end fraction