1.1 Prime factors, square roots and cube roots

Mathematics

Grade 8

The number system

Module 2

Prime factors, square roots and cube roots

CLASS ASSIGNMENT 1

1. Prime factors

• How do you write a number as the product of its prime factors?
• And how do you write it in exponent notation?

E.g. Question: Write 24 as the product of its prime factors(remember that prime factors are used as divisors only)

Prime factors of 24 = {2; 3}

24 as product of its prime factors: 24 = 2 x 2 x 2 x 3

24 = 2 ³ x 3 (exponential notation)

• Now express each of the following as the product of their prime factors(exponential notation) and also write the prime factors of each.

2. Square roots and cube roots

• How do you determine the square root ( $\sqrt{}$ )or cube root ( $\sqrt[3]{}$ )of a number with the help of prime factors?
• Do you recall this?

• Determine: $\sqrt{\text{324}}$ Step 1: break down into prime factors Step 2: write as product of prime factors (in exponential notation)Step 3: $\sqrt{\text{324}}$ means (324) ^½ (obtain half of each exponent)

Therefore: $\sqrt{\text{324}}$ = (2 ² x 3 ⁴ ) ^½ = 2 ¹ x 3 ² = 2 x 9 = 18

(324 is a perfect square, because 18 x 18 = 324)

• Remember: $\sqrt{}$ means (......) ^½ and $\sqrt[3]{}$ means (......) ^1/3

$\sqrt[3]{{\mathrm{8x}}^{\text{12}}}={\mathrm{2x}}^{\text{12}÷3=4}$ therefore ${\mathrm{2x}}^4$

2.1 Calculate with the help of prime factors:

(i) $\sqrt{\text{1 024}}$

(ii) $\sqrt[3]{\text{1000}}$

2.2 Calculate:

a) (2 x 3)² =

b) 3 x 8² =

c) $\sqrt[3]{1}$ =

d) $\sqrt[]{1}$ =

e) ${\left(\sqrt{2}\right)}^2$ =

f) then ${\left(\sqrt{\text{17}}\right)}^2$ =

g) (3 + 4) ³ + 14 =

h) $\sqrt{\text{36}}+\sqrt{9}$ =

i) $\sqrt{\text{36}+\text{64}}$ =

j) $\sqrt[3]{\text{27}}$ + $\sqrt[3]{1}$ =

k) $(\sqrt[3]{\text{27}}{)}^3$ =

l) $\sqrt{\text{64}{x}^{\text{12}}}$ =

HOMEWORK ASSIGNMENT 1

1. Determine the answers with the help of prime factors:

1.1 $\sqrt[3]{4\text{096}}$ 1.2 $\sqrt[4]{1\text{296}}$

2. Determine the answers without using a calculator.

2.1 $\sqrt[3]{3\text{.}3\text{.}3\text{.}3\text{.}{3}^2}$ =

2.2 $\sqrt[3]{5^3{a}^6{b}^{\text{15}}}$ =

2.3 $\sqrt[3]{8÷\text{125}\times \text{27}}$ =

2.4 $\sqrt[3]{\text{64}}+(\sqrt[3]{\text{64}}{)}^3$ =

2.5 $2(\sqrt[3]{8}{)}^3$ =

2.6 $\sqrt{\text{169}}$ =

2.7 $\sqrt{(6+4\times \text{12}{)}^2}$ =

2.8 $\sqrt{6\times 18\times \text{12}}$ =

2.9 $2(\sqrt{9}{)}^2$ =

2.10 $\sqrt{(6+3{)}^2}-3^3$ =

CLASS ASSIGNMENT 2

1. Give the meaning of the following in your own words (discuss it in your group)

• LCM :

Explain it with the help of an example

• BCD :

Explain it with the help of an example

2. How would you determine the LCM and BCD of the following numbers?

8; 12; 20

Step 1: Write each number as the product of its prime factors.(Preferably not in exponential notation)

8 = 2 x 2 x 2

12 = 2 x 2 x 3

20 = 2 x 2 x 5

Step 2: First determine the BCD (the number/s occurring in each of the three)Suggestion: If the 2 occurs in each of the three, circle the 2 in each number and write it down once), etc.

BCD = 2 x 2 = 4

Step 3: Now determine the LCM. First write down the BCD and then find the number that occurs in two of the numbers and write it down, finally writing what is left over)

LCM = 4 x 2 x 3 x 5 = 120

3. Do the same and determine the BCD and LCM of the following:

38; 57; 95

Calculate it here:

38 = ....................................................................

57 = ....................................................................

95 = ....................................................................

BCD = .................................. and LCM = ..................................

Assessment