Mr. Sardinha Math 10

1.4 Rational and Irrational Numbers (Estimating roots)

If a real number is not rational, it must be irrational. All numbers with square roots that are rational must be perfect squares. This is also true for cube roots.

For example: √25 This is a rational number since 25 is a perfect square; √25 = 5

√253

This is an irrational number since √253

is not a perfect cube. Remember, a rational number is a number that can be represented as a fraction. An irrational number is a non-repeating, or non-terminating decimal value. Therefore, when writing a value for an irrational number, it is just an approximation. Irrational numbers do not have to be the n th root of an integer. Here are some other examples of irrational

numbers: π, e, 1.010010001..., √2 The ability to estimate mentally the square root of a non-perfect square is important when checking a calculation for possible error.

Estimate the Square Root Mentally Example 1. Between what two consecutive integers are the following? ALSO, estimate the value mentally to one decimal place.

a) √46 b) √18

c) − √5 d) √√95

12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225

Approximating Irrational Numbers when given values of other irrational numbers With the use of a calculator, an approximation of the following values were obtained.

√7 ≈ 2.65 √70 ≈ 8.37 √700 ≈ 26.5 √7000 ≈ 83.7

Notice that √7 and √700 have the same numerals, but different decimal point answers. This is because

√700 = √7 x √100 = √7 x 10. The same is also true for √70 and √7000 = √70 x √100

Example: Given √17 ≈ 4.12 and √170 ≈ 13.04 , determine the value of the square root:

a) √1.7 ≈ b) √1700 ≈ c) √0.017 ≈

Cube Root

All numbers (positive and negative) have one cube root, denoted by the symbol √3

.

➢ The cube root of 1000 is 10. √10003

= 10

➢ The cube root of –27 is –3. √−273

= −3 Estimate the Cube Root Mentally Example 2. Between what two consecutive integers are the following? ALSO, estimate the value mentally to one decimal place.

a) √113

b) √1203

Other Roots

4 x 4 =16 so a square root of 16 is 4 or √16 = 4

5 x 5 x 5 = 125 so the cube root of 125 is 5 or √1253

= 5

2 x 2 x 2 x 2 = 16 so a fourth root of 16 is 2 or √164

= 2

13 = 1 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 73 = 343 83 = 512 93 = 729 103 = 1000

Note:

3√2 ≠ √23

4√2 ≠ √24

Radicals - Any expression of the form √𝑥𝑛

is called a radical.

➢ The index is the number of times the radical must be multiplied by itself to equal the radicand. ➢ If the index in a radical is even, then the radicand must be positive. ➢ When the index is not written in the radical, it is assumed to be 2.

Identify Index and Radicand Example 3. Identify the index and the radicand in each of the following.

a) √755

b) √50 c) √−1

10

3

Evaluate the Root Mentally Example 4. Mentally evaluate, where possible.

a) √49 b) √−643

c) √100004

d) √1

32

5 e) √−16

4 f) √125

3

Evaluate the Root Using a Scientific Calculator

Example 5. Use a calculator to evaluate to the nearest hundredth. (Use √𝑥𝑛

button)

a) √10245

b) √−21877

c) −√506254

d) √1255

e) √0.56

f) √10004

Classwork: Mickelson pp.31-34, #1 – 8