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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.

Image

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.

Example 1: Rational exponents

Question

Write each of the following as a radical and simplify where possible:

Answer

  1. 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 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

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:











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