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A power is an algebraic or numeric expression that consists of a base and an exponent. In this lesson, exponential notation will be extended to rational numbers.

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.

Calculate the value of the following expressions. Write the answer as an integer number or as a fraction in its simplest form.

e Use the Power of a Power Property to simplify the expression

The Meaning of Exponent

The exponent of an expression indicates how many times the is multiplied by itself.

Power Multiplication Value
This is perfectly defined when the exponent is a natural number. However, what happens if an exponent is a rational number? For example, what is the value of

Connection Between Rational Exponents and Radicals

By definition, to calculate the value of a power, the base is multiplied by itself as many times as indicated by the exponent. So, what happens with expressions where the exponent is a rational number, such as It might be confusing to multiply by itself times.
Considering the Power of a Power Property, notice what happens when the expression is raised to the third power.
If the Power of a Power Property holds true for rational exponents, then must be a number that, when raised to the third power, equals Such a number already exists:
Therefore, for this property to still work, raised to the power of is defined as the cube root of

Definition of

The relationship between rational exponents and roots can be extended to any rational exponent of the form where is a natural number.



For any real number and natural number the expression is defined as the root of Note that a root with an even index is defined only for non-negative numbers. Therefore, if is even, then must be non-negative.
power of 1/n as root
With this definition, any power with an exponent of the form can be written as a radical.

Rewriting a Power as a Radical

Dominika pays dollars per hour to study with a private tutor.

Dominika's homework

As her first homework assignment, she was given the task of writing how much she pays per hour as a radical. Write her answer without including the currency symbol.

Use the radical expression to calculate the value of


Raising any positive number to the power of is equivalent to calculating the fourth root of the number.


To calculate the value of rewrite it as a radical using the definition of

Pop Quiz

Calculating Powers Using Radicals

Calculate the values of the given powers with rational exponents by using a radical.

The applet that generates powers with rational exponents

Finding a Radical and Rewriting It as a Power

YBC is an ancient Babylonian clay tablet believed to be the work of a student who lived in southern Mesopotamia around the year The tablet contains an extremely accurate approximation of the length of the diagonal of a square with side length
Unit square with a diagonal
The length of the diagonal is given to an accuracy of six decimal digits, the greatest known computational accuracy in the ancient world. Using the Pythagorean Theorem, calculate the exact length of the diagonal. Give the answer as a radical.
Write the length of the diagonal using a rational exponent.


Consider a right triangle in which the legs are two sides of the square and the hypotenuse is the diagonal.


Consider the right triangle formed by two sides of the square and its diagonal.

A right triangle which is a half of the unit square
Here, the length of both legs is Therefore, by substituting and into the Pythagorean Theorem, the length of the diagonal can be found.
Solve for
The length of the diagonal, written as a radical, is exactly units. This number can be also expressed as a power with a rational exponent of

Definition of a Rational Exponent

Note that the numerator of a rational number does not have to be but could also be any integer number. Therefore, rational numbers with a numerator different than should also be considered as rational exponents.


Rational Exponent

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.
Rational exponent to root
Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which is raised. Since is a denominator, it cannot be zero. Moreover, if is an even number, then must be non-negative.

Another Radical Equivalent to

Using the Power of a Power Property, another expression equivalent to involving a radical can be found.
Therefore, can also be defined as Note that if is an even number, then must be non-negative. In conclusion, there are two definitions for

Expressing a Rational Exponent as a Radical

Ignacio ate some cake at a birthday party. When he arrived home, he told his parents that he had slices.

Ignacio telling his parents how many slices of the cake he ate
His mother, who is a math teacher, asked him to express the number of slices that he ate as a radical.
His father, who does not understand powers at all, asked him to calculate the value of this exponential expression.


Raising any number to the power of is equivalent to calculating the cube root of the square of the number.


To calculate the value of it should be first rewritten as a radical. In order to do that, the definition of a rational exponent can be used.

Now Ignacio is able to give an answer for his mother. He can say that he ate slices of the cake. To answer his father, Ignacio needs to find the value of the found radical.

Principal Root

Now, focus on roots with an even index. For example, consider By definition, is a number that, when raised to the second power, equals
What about all the numbers that are equal to when raised to the second power? There are two such numbers, and
To avoid complicating the definitions of and positive is conventionally defined as the principal root. Therefore, for any even number is defined as the positive number that, when raised to the power, equals

Translating Between Radicals and Rational Exponents

It is important to know how to write expressions with rational exponents as radicals. Sometimes it is required to simplify an expression by using only radicals. Consider the following example.


Start by writing both numeric expressions using radicals.


Start by writing both factors in the expression using radicals.

Moreover, it is important to know how to write radical expressions using rational exponents, since rational and integer exponents have the same properties. Simplifying expressions involving exponents may be much easier than simplifying expressions involving radicals. Consider one last example.


Write using a rational exponent and then use the Product of Powers Property.


The expression can be simplified by writing using a rational exponent and then using the Product of Powers Property.
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