Estimating Surds
Estimating Surds
If the \(n^{\text{th}}\) root of a number cannot be simplified to a rational number, we call it a surd. For example, \(\sqrt{2}\) and \(\sqrt[3]{6}\) are surds, but \(\sqrt{4}\) is not a surd because it can be simplified to the rational number \(\text{2}\).
In this tutorial we will look at surds of the form \(\sqrt[n]{a}\) where \(a\) is any positive number, for example, \(\sqrt{7}\) or \(\sqrt[3]{5}\). It is very common for \(n\) to be \(\text{2.}\) so we usually do not write \(\sqrt[2]{a}\). Instead we write the surd as just \(\sqrt{a}\)
It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to estimate where a surd like \(\sqrt{3}\) is on the number line. From a calculator we know that \(\sqrt{3}\) is equal to \(\text{1.73205...}\). It is easy to see that \(\sqrt{3}\) is above \(\text{1}\) and below \(\text{2}\). But to see this for other surds like \(\sqrt{18}\), without using a calculator you must first understand the following:
If \(a\) and \(b\) are positive whole numbers, and \(a
A perfect square is the number obtained when an integer is squared. For example, \(\text{9}\) is a perfect square since \({3}^{2}=9\).
Similarly, a perfect cube is a number which is the cube of an integer. For example, \(\text{27}\) is a perfect cube, because \({3}^{3}=27\).
Consider the surd \(\sqrt[3]{52}\). It lies somewhere between \(\text{3}\) and \(\text{4.}\) because \(\sqrt[3]{27}=3\) and \(\sqrt[3]{64}=4\) and \(\text{52}\) is between \(\text{27}\) and \(\text{64}\).
The following video explains how to estimate a surd.
Example
Question
Find two consecutive integers such that \(\sqrt{26}\) lies between them. (Remember that consecutive integers are two integers that follow one another on the number line, for example, \(\text{5}\) and \(\text{6}\) or \(\text{8}\) and \(\text{9}\).)
Use perfect squares to estimate the lower integer
\({5}^{2}=25\). Therefore \(5<\sqrt{26}\).
Use perfect squares to estimate the upper integer
\({6}^{2}=36\). Therefore \(\sqrt{26}<6\).
Write the final answer
\(5<\sqrt{26}<6\)
Example
Question
Find two consecutive integers such that \(\sqrt[3]{49}\) lies between them.
Use perfect cubes to estimate the lower integer
\({3}^{3}=27\), therefore \(3<\sqrt[3]{49}\).
Use perfect cubes to estimate the upper integer
\({4}^{3}=64\), therefore \(\sqrt[3]{49}<4\).
Write the answer
\(3<\sqrt[3]{49}<4\)
Check the answer by cubing all terms in the inequality and then simplify
\(27<49<64\). This is true, so \(\sqrt[3]{49}\) lies between \(\text{3}\) and \(\text{4}\).
This lesson is part of:
Algebraic Expressions Overview