ROOT AND POWER

When a quantity m is raised to power in. It means that the quantity m should be multiple in the number of times of n 

Mⁿ = m₁ x m₂ x m₃...mₙ

2³ = 2 x2 x2 

From the above examples we can see that power gives the number of times the base is to be used as a factor. Power is also known to be an index or exponent.

Base -------X ʸ⁻⁻⁻⁻⁻⁻ power ( index)

8² mean 8 is raised to the power of 2 or 8 required 

z³ = means z cubed

2⁵  means fifth power of two

*Rule of  powers*

1. When a negative base is raised to an even power. it gives a positive result.

a.  (-2)² = (-2) x (-2)      =+4   

b. (-3)⁴ = (-3) x (-3) x (-3) x (-3) = +81

ii. when a negative base is raised to an odd number, it gives a negative result 

a.  (-2)³ = (-2) x (-2) x (-2) = -8

b. (-3)⁵ = (-3) x (-3) x (-3) x (-3)x (-3) = -243

iii. When a positive base is raiaed to either even or odd number, the result is positive.

a.  (-2)³ = (-2) x (-2) x (-2) = -8

b. (-3)⁵ = (-3) x (-3) x (-3) x (-3)x (-3) = -243

iii. When a positive base is raiaed to either even or odd number, the result is positive.

a. 2³ = 2 x 2 x 2 = 8

b.  3⁴ = 3 x 3 x3 x3 = 

     81

When any base ( except zero ) is raised to zero power , the result is positive one

a. x⁰ = 1

b. (-10)⁰ = 1

note 0⁰  = 1

When a base is raised to a negative power, the result is reciprocal of the base of the  base of the original number raised to the positive of that  power

a⁻ˣ = 1/aˣ

  *Worked examples*

Evaluate the following

I.   4² = 4 × 4 = 16 

II.  (-4)² = -4 × -4 = 16

III. 2² x (-3)³

  = -2 x -2 x(-3)(-3)(-3)

= -4 x (-27)

= 108

iv. (3)⁻² x (-2)3

2. Given that x=3, y= -1, and z = -2. Evaluate

x³ + y³ + z³

(3)³ + (-1)³ + (-2)³

27 + (-1) + (-8)

 = 18

ROOT

A root is obtained when a base is raised to a positive fraction power. The fractional power forms. Study the below question carefully

WORKED EXAMPLES

Find the value of y in each of the following 

Z^1/n  = ⁿ√(Z)

Where ( ^) means  ofpower of

1. 8^⅓ = ³√(8) 

= ³√(2 x 2 x 2) 

= 2

Example 2

What is the value of 

y⁴ = 81

add cube root to both side

⁴√(y⁴) = ⁴√(81)

Note: its only a fourth root that can cancel fourth power of y .We need to establish (y) alone

Hence,

y = ⁴√(81)

y = ⁴√(3 x 3 x3 x3)

y = ⁴√(3)⁴

y = 3

now you can see that its the fractional power of a number that forms its root and that the root of a number equals (=) to  one of its equal factors. 

*Rules of roots*

Given that y = ⁿ√(z)

a.   If n is a even number and y > 0, y will have two real solution, but if n is odd, y will have only two solution

Exampe 3

y² = 9 

add squared roots to both side

and so y = 

√(9) = + 3 or -3

4.   y³ = 64 

add cube root to both side

³√(y³) = ³√(64)

y = ³√(4 x 4 x4)

y = ³√(4³)

and so y 

y = 4

b. If n is an even number and y <0, z will have no real root, but if n is an odd number and y < 0, z will have only one z (negative) root

Example 5

a. y² = -4, and so y = √(-4) = No real root

b. y³ = - 8, 

and so x

 = ³√(8) =

- 2

Example 7

i √(7056)

 √(16 x 441)

√(16) x √(441)

4 x 21 = 84

ii. √(14400)

√(441 x 100)

= √(441) x √(100)

= 12 x 10 =120

iii. ³√(-2744)

³√(-8 x 343)

= ³√(-8) x ³√(343)

     -2 x  7 = -14

Example 8

if p = -7 and q = -2, find

a. (p-q)(p+q)

[(-7)-(-3)][(-7) + (-2)]

(-7 +3)(-7 - 2)

(-4)(-9)

=36

 

b.  p² - 2pq + q²

= (p -q)(p-q)

  [ (-7)-(-2)][(-7)-(-2)]

 

(-7 +2)(-7 +2)

= (-5)(-5) = 25

c. p + 2q - p²

(-7) + (2 x -2) - (-7)²

-7 + (-4) - (49)

-7 -4 -49

-60

SEE LINEAR INEQUALITIES

LINEAR INEQUALITY

Inequality had been part of life when two things differs in content, values privilege and quantity when it should be on the same ground State. The phenomenon called INEQUALITY had been set in place

Inequalities means ordering. The order or position of 2 in relation to 3 or 0, in the real number system, can be described as follows: 2 is less than 3 and 2 is greater than 0.

The symbol '<' is used to mean 'strictly less than' , and the symbol '>' means 'strictly greater than'. The symbol '≥' means 'greater than or equal to' while the symbol '≤' means 'less than or equal to'.

PROPERTIES OF INEQUALITIES

If a, b and c are real numbers then the following relation hold:

1. a < b⇔b - a > 0 , where the symbol '⇔' means the expression on the right is true if the previous statement holds.

2. One and only one of a < b, a = b , a > b is always true

3. If a > 0 and b > 0 , then a + b >0

4. If a >0 and b > 0 , then ab > 0

5. If a < b , there exists a 'c' such that a < c < b

6. If 'a' is given , then we can find some b and c such that b < a < c.

Example 1: If a, b and c ∈ R and a < b , where R is the set of real numbers, prove that

(i) a + c < b + c

(ii) a - c < b - c

(iii) ac < bc if c > 0

(iv) ac > bc if c < 0

Solution

(i) a < b ⇔ b - a > 0 by property 1 

But ( b + c ) - ( a + c ) = b - a for any c

( b + c ) - ( a + c ) > 0

b + c > a + c

An inequality is preserved if a fixed quantity is added to each side of it

(ii) Since a < b ⇒ b - a > 0, the symbol '⇒' is read as 'implies that;

     Then ( b - c ) - ( a - c ) = b - a

    ( b - c ) - ( a - c ) > 0

    b - c > a - c

An inequality is preserved if a fixed quantity is subtracted from each side of it.

(iii) Since a < b ==> b - a > 0

      Then for any c > 0, ( b - a )c > 0

      bc - ac > 0

     bc > ac

Multiplying each side of an inequality by a fixed positive quantity doesn't alter the inequality.

(iv) If a < b and c < 0 then

      b - a > 0 and ( -c ) > 0

     ( b - a )( -c ) > 0

     -bc + ac > 0

     ac > bc

Multiplying each side of an inequality by a fixed negative quantity reverses the inequality.

Example 2: Find the range of values of x for which each of the following will hold:

(i) ¾x - 2 < 4

(ii) ⅔( x - 1 ) > ½( 1 - 2x ) + ½

Solution

(i) ¾x - 2 < 4

     Multiplying both sides by 4, gives 3x - 8 < 16

     ⇔ 3x - 8 + 8 < 16 + 8

      ⇔ 3x < 24

           x < 8

(ii) ⅔( x - 1 ) > ½( 1 - 2x ) + ½

      Multiplying both sides by 6

     4( x -1 ) > 3( 1 - 2x ) + 3

    4x - 4 > 3 - 6x + 3

    10x > 10

     x > 1

Example 3: Solve the inequality 4x - 3 ≥ 2x + 3

Solution

4x - 3 ≥ 2x + 3

⇔ 4x - 2x - 3 + 3 ≥ 2x - 2x + 3 + 3

 ⇔ 2x ≥ 6 or x ≥ 3