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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
where , and , .

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 n root of a number. For example, and , so both −2 and 2 are square roots of 4.

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.

Exercise 1: Rational exponents and surds

Simplify the following and write answers with positive exponents:


Use the laws to re-write 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


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 .

Sometimes a surd cannot be simplified. For example, are already in their simplest form.

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.

Exercise 3: Rationalising the denominator

Rationalise the denominator in each of the following:

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