Everything Maths Grade 10

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

p

52. It lies somewhere between 3 and 4, because^3

p

27 = 3and^3

p
64 = 4and 52 is between
27 and 64.

VISIT:

The following video explains how to estimate a surd.

See video:2DDVatwww.everythingmaths.co.za

Worked example 5: Estimating surds

QUESTION

Find two consecutive integers such that

p

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.)

SOLUTION

Step 1: Use perfect squares to estimate the lower integer

52 = 25. Therefore 5 <

p

26.

Step 2: Use perfect squares to estimate the upper integer

62 = 36. Therefore

p

26 < 6.

Step 3: Write the final answer

5 <

p

26 < 6

Worked example 6: Estimating surds

QUESTION

Find two consecutive integers such that^3

p

49 lies between them.

SOLUTION

Step 1: Use perfect cubes to estimate the lower integer

33 = 27, therefore 3 <^3

p

49.

Step 2: Use perfect cubes to estimate the upper integer

43 = 64, therefore^3

p

49 < 4.

Step 3: Write the answer

3 <^3

p

49 < 4

Step 4: Check the answer by cubing all terms in the inequality and then simplify

27 < 49 < 64. This is true, so^3

p

49 lies between 3 and 4.

Chapter 1. Algebraic expressions 15