Assignment: Module 8 Index Laws, Algebraic Fractions and Equations

This assignment solution addresses key concepts from Module 8 on Index Laws. It begins by defining index notation, identifying the base, index, and basic numeral in exponential equations. The solution then explains the meaning of the index. The assignment proceeds to the application of index laws, including simplification of expressions using given index laws and completing operations with algebraic fractions, such as multiplication and division, using general rules. Finally, the solution provides examples of solving equations where the unknown is part of the index and demonstrates how to solve exponential equations with different bases.

1)
a)
2 is the base, 3 is the index and 8 is the basic numeral
b)
Index part means the power to which the base is raised up to.
2)
c)
a1=a
a−n= 1
an
d)
Expression Index Law Equivalent Expression
i ( xyz )5 (a ¿¿ m)n=amn ¿ x5 y5 z5
ii x2 x3 x−4 am an=am+n x
iii y3
y−6
am
an =am−n y9
iv (x ¿¿−5)2 ¿ (a ¿¿ m)n=amn ¿ x−10
v
( y4
x−2 )
3 (a ¿¿ m)n=amn ¿
( y12
x−6 )
e)
( a2 b3
a7 b−8 )
−2
= ( a−4 b−6
a7 b−8 ) =a−11 b2
3)
g)
a
b ÷ c
d = ad
bc
a
b − c
d = ad−bc
bd
a
b + c
d = ad +bc
bd
h)

2
10 a2−5 a ÷ 9 a
8 a−4 = 2(8 a−4)
9 a(10 a2−5 a)
2
a+4 + 3
a−1 = 2(a−1)+3 (a+4)
(a+ 4)(a−1) = 5 a+10
(a+ 4)(a−1)
4)
i)
x=5
y=0.5
j)
To solve equations where the unknown is part of the index… First write all terms with the same base
, then combine terms on each side of the equation and equate the indices
k)
16x 643 x−2=43
42 x 43(3 x−2)=43
42 x 49 x−6=43
411 x−6=43
11 x−6=3
x= 9
11