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When solving geometry problems, radical expressions — like square roots or cube roots — are present in a lot of concepts and formulas. For example, the diagonal of a square is calculated by where is the side length of the square. This lesson will explore the operations on more general radical expressions and how radicals might apply to geometry and physics.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Here are a few practice exercises before getting started with this lesson.

a Write as a radical.
b Consider the following algebraic expression. Assume that the variable is positive.
What is the simplest form of the expression?
c Write the following expression in its simplest form.
Challenge

Length of a Triangular Fencing

Ignacio is planning to build an astronomical observatory in his garden. The building will be enclosed by a fence with a triangular shape. The dimensions of Ignacio's garden are presented in the following diagram.

Ignacio's garden with a fencing

In this diagram, all dimensions are measured in meters. Although some side lengths are still not decided, help Ignacio calculate the length of the fence with respect to What is the value of

Discussion

Root

The root of a real number expresses another real number that, when multiplied by itself times, will result in In addition to the radical symbol, the notation is made up of the radicand and the index
The resulting number is commonly called a radical. For example, the radical expression is the fourth root of Notice that simplifies to because multiplied by itself times equals
The general expression represents a number which equals when multiplied by itself times.

Some numbers have more than one real root. For example, has two fourth roots, and because both and are equal to The number of real roots depends on the sign of a radicand and an integer

is Even is Odd
Two unique real roots, and One real root,
One real root, One real root,
No real roots One real root,

By the definition of an root, calculating the power of the root of a number results in the same number The following formula shows what happens if these two operations are swapped.

Rule

Roots of Powers

To simplify an root, the radicand must first be expressed as a power. If the index of the radical and the power of the radicand are equal such that the radical expression can be simplified as follows.

The absolute value of a number is always non-negative, so when is even, the result will always be non-negative. Consider a few example roots that can be simplified by using the formula.

Is even? Simplify

Proof

Informal Justification

To write the expression for there are two cases to consider.

Both cases will be considered one at a time.

Start by noting that since is non-negative, is also non-negative. This means that the root can be rewritten using a rational exponent.
Since both and are rational numbers, the Power of a Power Property for Rational Exponents can be applied to simplify the obtained expression.
Simplify

Therefore, is equal to if is non-negative.

In case of a negative value of there are also two cases two consider.

  • is even
  • is odd

Even

Recall that a root with an even index is defined only for non-negative numbers. Although is negative, is positive. Also, a power with a negative base and an even exponent can be rewritten as a power with a positive base.
Now, since is positive, the Power of a Power Property for Rational Exponents can be applied again to simplify
Simplify

Odd

A root with an odd index is defined for all real numbers. By the definition of the root, the expression is the number that, when multiplied by itself times, will result in
Because is odd and is negative, is also negative. This means that the best candidate for is simply

Summary

If is non-negative, is always equal to However, in case of negative the value of depends on the parity of

Even
Odd

To conclude, for odd values of the expression is equal to On the other hand, if is even, can be written as

Extra

Simplifying

Depending on the index of the root and the power in the radicand, simplifying may be problematic. Because real roots with an even index are defined only for non-negative numbers, the absolute value is sometimes needed.

If is even, is defined only for non-negative

The following applet presents a decision tree to simplify In this applet, is the greatest common factor of and
Simplifying nth root of a to the mth power
Consider example roots that can be simplified by using the decision tree.
Simplify
Example

Homework on Radical Expressions

In addition to physics and astronomy, Ignacio is also interested in algebra. Unfortunately, he missed one class and he cannot solve his homework on radical expressions. Help Ignacio pair each expression to its simplified form.

Hint

Write the radicand as a power. Then, use the Roots of Powers Property to simplify the radical expression.

Solution

To simplify the given expressions, the Roots of Powers Property will be used.
Unfortunately, only the second expression is written in the form that allows to use the property right away. Because the power and the index of the root are both and is an odd number, the second expression simplifies to
Radical Expression Simplified Form
Each of the remaining expressions should be rewritten as a power with the exponent equal to the index of the corresponding root. Notice that the radicand in the first expression is a perfect square trinomial.
Now, the radical expression can be simplified by writing the radicand as the square of a binomial. Because the index of the root is an even number, the result will require the absolute value.
Therefore, the first expression simplifies to
Radical Expression Simplified Form
Next, properties of exponents will be used to write the radicand as a power with the exponent of
Rewrite

Since is non-negative, is also non-negative. This means that is equal to
Radical Expression Simplified Form
Now the final expression will be examined. Similar to the previous expression, properties of exponents will be used to rewrite the radicand as a square.
Simplify
This time, it is not known whether is non-negative. Therefore, the absolute value is needed in this expression. Finally, all simplified forms may be written in the table.
Radical Expression Simplified Form
Discussion

Product and Quotient Properties of Radicals

Usually, the Roots of Powers Property is not enough to simplify radical expressions. Therefore, more properties will be presented and proven in this lesson. The first one refers to the root of a product.

Rule

Product Property of Radicals

Given two non-negative numbers and the root of their product equals the product of the root of each number.

for and

If is an odd number, the root of a negative number is defined. In this case, the Product Property of Radicals for negative and is also true.

Proof

First, a special case of the property will be proven. Assume that or is equal to By the Zero Property of Multiplication, the radicand in left-hand side of the formula is equal to
Because root of is equal to This means that both the left-hand side and one of the factors on the right-hand side equal Again, by the Zero Property of Multiplication, the right-hand side is also
Therefore, the property is true when or is equal to Next, assume that both and are nonzero. Let and be real numbers such that and By the definition of an root, each of these numbers raised to the power is equal to its corresponding radicand.
Next, because is not equal to Equation (I) can be multiplied by which is equal to
Now, substitute Equations (II) and (III) into the obtained equation.
Substitute values and simplify
Since and are of the same sign, the final equation implies that