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Radical Functions

Rational Exponents and Radicals

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Rational exponents and radicals are two different ways to express the same quantities. Sometimes, one form is more useful than the other. Thus, being able to translate between the two is important.

Rule

Rational Exponents

When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.
Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which a is raised. The Properties of Exponents apply to rational exponents in the same way they apply to integers. Consider
Thus, is equivalent to

Example

Rewrite between rational exponent form and radical form

fullscreen
Rewrite the given expressions in the opposite form.
Show Solution expand_more

Example

To begin, notice that the expression is written in radical form. Thus, it needs to be rewritten with a rational exponent. Recall that Notice that in x does not have an exponent. That means it is raised to the power of 1. This gives

Example

This expression has a rational exponent. Thus, we must rewrite it with a radical. Recall that Since the denominator of the exponent is 3, we can write the cube root of x2.

Theory

Properties of Radicals

The properties of radicals allow expressions with radicals to be rewritten.

Rule

Product Property of Radicals

Rule

The nth root of a product can be written as the product of the nth root of each factor. For example, can be expressed as follows using this rule.
This rule can be explained by expressing the radical using a rational exponent, and then using the Power of a Product Property.
This rule is valid for all values of a, b and n. However, in order to avoid non-real solutions it is necessary that a0 and b0 for even values of n.

Rule

Quotient Property of Radicals

Rule

The nth root of a fraction can be written as the nth root of the numerator divided by the nth root of the denominator. For example, can be expressed as follows using this rule.
This rule can be explained by rewriting the radical as a rational exponent, and then using the Power of a Quotient Property.
This rule is valid for all values of a and n, and when b0. However, in order to avoid non-real solutions it is necessary that a0 and b>0 for even values of n.

Example

Simplify the expression using the properties of radicals

fullscreen
Simplify the expression using the properties of radicals.
Show Solution expand_more
When two radicals with the same index are either multiplied or divided they can be simplified using the properties of radicals. We'll begin by simplifying the numerator using the Product Property of Radicals.
Next we simplify the expression using the Quotient Property of Radicals.
3
Thus, the expression simplifies to 3.

Rule

Roots of Powers

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

The absolute value of a number is always non-negative, so when n is even, the result will always be non-negative.

Proof

Informal Justification

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

  • a0
  • a<0

Both cases will be considered one at a time.

a0

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

a1
a
Therefore, is equal to a if a is non-negative.

a<0

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

  • n is even
  • n is odd

Even n

Recall that a root with an even index is defined only for non-negative numbers. Although a 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 -a is positive, the Power of a Power Property for Rational Exponents can be applied again to simplify
Simplify

(-a)1
-a

Odd n

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

Summary

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

a0 a<0
Even n
Odd n

To conclude, for odd n, the expression is equal to a. On the other hand, if n is even, can be written as a.

Method

Simplifying Radical Expressions

Expressions with radicals can be written using rational exponents. Then, they can be simplified using the properties of exponents. Consider the following expression.
1
Rewrite terms into
expand_more
When a term inside a radical has a power greater than the index of the radical, it can be rewritten into -form. In the example, there are two such terms, and
First, the Product of Powers Property can be used to rewrite the terms under these radicals.
The products under the radicals can now be rewritten using the Product Property of Radicals.
2
Simplify -terms
expand_more
The -terms can now be simplified as follows.
In the expression there are two terms that can be simplified using this rule.

3
Express the radicals using rational exponents
expand_more
Next, rewrite the radicals using rational exponents. The example can be rewritten as follows.
4
Simplify using the laws of exponents
expand_more
When the expression is written using rational exponents, it can be simplified. When two terms with the same base are multiplied, the exponents are added according to the Product of Powers Property.

a=a1

To simplify the exponent further, which requires adding and subtracting fractions, the denominators must be made the same. Here, the least common denominator is 12.
Since the expressions in the numerator and the denominator have the same bases, x, they can be simplified. First, by using the Quotient of Powers Property, the expression is written as one term. To simplify the exponent, the denominators must then be made the same.
5
Rewrite expression using radicals
expand_more
When the expression has been simplified completely, it can be rewritten using radicals.
This is the simplified expression.

Example

Simplify the expression with rational exponents

fullscreen
Simplify the expression using the properties of exponents.
Show Solution expand_more
Notice that the terms in this expression each have a difference base.
This expression cannot be simplified until the bases are the same. Therefore, we can begin by rewriting the terms with base 2. Notice that 4=22 and 16=24.
Next, we can rewrite the expression using the Power of a Power Property.
All terms in the expression now have the same base. From here, we can use the Product of Powers Property and the Quotient of Powers Property to simplify.
The answer is which can also be written as
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