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
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