Rational exponents and surds
The laws of exponents can also be extended to include the rational numbers. A rational number is any number that can be written as a fraction with an integer in the numerator and in the denominator. We also have the following definitions for working with rational exponents.
Identity 1
 If , then
For , we say that 5 is the square root of 25 and for , we say that 2 is the cube root of 8. For , we say that 2 is the fifth root of 32.
When dealing with exponents, a root refers to a number that is repeatedly multiplied by itself a certain number of times to get another number. A radical refers to a number written as shown below.
The radical symbol and degree show which root is being determined. The radicand is the number under the radical symbol.

If is an even natural number, then the radicand must be positive, otherwise the roots are not real. For example, since , but the roots of are not real since .

If is an odd natural number, then the radicand can be positive or negative. For example, since and we can also determine since .
It is also possible for there to be more than one
A surd is a radical which results in an irrational number. Irrational numbers are numbers that cannot be written as a fraction with the numerator and the denominator as integers. For example, , , are surds.
Example 1: Rational exponents
Question
Write each of the following as a radical and simplify where possible:
Answer
 not real
Example 2: Rational exponents
Question
Simplify without using a calculator:
Answer
Write the fraction with positive exponents in the denominator
Simplify the denominator
Take the square root
Exercise 1: Rational exponents and surds
Simplify the following and write answers with positive exponents:
Simplify:
Use the laws to rewrite the following expression as a power of :
Simplification of surds
We have seen in previous examples and exercises that rational exponents are closely related to surds. It is often useful to write a surd in exponential notation as it allows us to use the exponential laws.
The additional laws listed below make simplifying surds easier:
Identity 2
Example 3: Simplifying surds
Question
Show that:
Answer
Examples:
Like and unlike surds
Two surds and are like surds if , otherwise they are called unlike surds. For example, and are like surds because . Examples of unlike surds are since .
Simplest surd form
We can sometimes simplify surds by writing the radicand as a product of factors that can be further simplified using .
Example 4: Simplest surd form
Question
Write the following in simplest surd form:
Answer
Write the radicand as a product of prime factors
Simplify using
Sometimes a surd cannot be simplified. For example, are already in their simplest form.
Example 5: Simplest surd form
Question
Write the following in simplest surd form:
Answer
Write the radicand as a product of prime factors
Simplify using
Example 6: Simplest surd form
Question
Simplify:
Answer
Write the radicands as a product of prime factors
Simplify using
Simplify and write the final answer
Example 7: Simplest surd form
Question
Simplify:
Answer
Factorise the radicands were possible
Simplify using
Simplify and write the final answer
Example 8: Simplest surd form with fractions
Question
Write in simplest surd form:
Answer
Factorise the radicands were possible
Simplify using
Simplify and write the final answer
Exercise 2: Simplification of surds
Simplify the following and write answers with positive exponents:
Simplify the following:
Rationalising denominators
It is often easier to work with fractions that have rational denominators instead of surd denominators. By rationalising the denominator, we convert a fraction with a surd in the denominator to a fraction that has a rational denominator.
Example 9: Rationalising the denominator
Question
Rationalise the denominator:
Answer
Multiply the fraction by
Notice that , so the value of the fraction has not been changed.
Simplify the denominator
The term in the denominator has changed from a surd to a rational number. Expressing the surd in the numerator is the preferred way of writing expressions.
Example 10: Rationalising the denominator
Question
Write the following with a rational denominator:
Answer
Multiply the fraction by
To eliminate the surd from the denominator, we must multiply the fraction by an expression that will result in a difference of two squares in the denominator.
Simplify the denominator
Exercise 3: Rationalising the denominator
Rationalise the denominator in each of the following: