ESTIMATING SURDS

If the nTH root of a number cannot be simplified to a rational number, we call it a surd. For example, 2√ and 6√3 are surds, but 4√ is not a surd because it can be simplified to the rational number 2.

In this chapter we will look at surds of the form a√n where a is any positive number, for example, 7√ or 5√3. It is very common for n to be 2, so we usually do not write a√2. Instead we write the surd as just 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 3√ is on the number line. From a calculator we know that 3√ is equal to 1,73205…. It is easy to see that 3√ is above 1 and below 2. But to see this for other surds like 18√ without using a calculator, you must first understand the following:

Identity 1

If a and b are positive whole numbers, and a<b, then a√n<b√n.
A perfect square is the number obtained when an integer is squared. For example, 9 is a perfect square since 32=9.

Similarly, a perfect cube is a number which is the cube of an integer. For example, 27 is a perfect cube, because 33=27.

Consider the surd 3√52. It lies somewhere between 3 and 4, because 3√27 = 3 and 3√64 = 4 and 52 is between 27 and 64.

Estimating Surds: Example 1

Question

Find the two consecutive integers such that √26 lies between them. (Remember that consecutive integers are two integers that follow one another on the number line, for example, 5 and 6 or 8 and 9).

Answer

Use perfect squares to estimate the lower integer 52 = 25. Therefore 5 < √26 .

Use perfect squares to estimate the upper integer 62 = 36. Therefore √26 < 6.

Write the final answer 5 < √26 < 6.

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