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```Per
Year 10A
Module 1
Indices
Lesson 1: Index Laws
Homework: 3F (p. 186): # 4, 6, 8, 11 ;
3G (p. 186): # 1-3, 5, 9, 10
Lesson 2: Rational (Fractional) Indices
Materials
Essential mathematics
3.6 - 3.9
Homework: 3I (p. 200): # 5, 8-10
Lesson 3: Scientific Notation
Homework: 3H (p. 195): # 4-6
Time Required
3 periods
Test date:________________l
Indices
1
Part 1: Indices Laws:
Things to Know:
Law 1: x a  x b  x a b
Describe in words:
How it’s done:
42 × 45 = 42 + 5
When multiplying terms with the same base, you add the
indices.
The bases do not change.
7
= 4
Simplify 3k2 × 4k8.
3k2 × 4k8 = 12k10
When multiplying pronumerals (or variables) with coefficients:


Calculate the product of the coefficients.
Check that the bases are the same.
If the bases are the same, add the indices.
You try:
Indices
2
Things to Know:
Law 2: x a  xb  x a b
Describe in words:
How it’s done:
Simplify 612 ÷ 63.
612 ÷ 63 = 6 12 – 3
= 69
Simplify 15p10 ÷ 3p.
In divisions with terms having the same base, subtract the index
of the divisor (second index).
Be careful: This only works when the bases are the same.
Remember that when the index is 1 it can be omitted.
15p10 ÷ 3p = 5p10 ÷ p1 = 5p10 – 1 = 5p9
You try:
Simplify:
You try:
Simplify:
Indices
3
Things to Know:
Law 3: x 0  1
Describe in words:
How it’s done:
Calculate the value of each of these expressions.
1. y0 = 1
2. 5y0= 5 × 1= 5
3. (5y)0 = 1
4. y0 + 70= 1 + 1= 2
The order of operation is very important in all
calculations:




brackets
powers
multiplication & division
Any number to the power of zero = 1
(Remember the debate about 00 which might be
undefined or might be 1 depending on how you
look at it.)
You try:
Simplify:
Indices
4
Things to Know:
Law 4: ( x a )b  x ab
Describe in words:
How it’s done:
Write (73)2 as an equivalent expression in simple
index form.
(73)2 = 7 3×2 = 7 6
Expand (h5)4.
(h5)4 = h5 × 4 =h9
Expand (3y7)3.
(3y7)3 = 33 × y7 × 3 = 27y21
When raising an expression in index form to another
power, multiply the indices and leave the base
unchanged.
Wherever possible, determine a method to check your
solutions.
Raise both factors inside the brackets to the power 3:
the coefficient, 3, and the power of the pronumeral, y7.
33 = 27 = (y7)3 = y7 × 3
When raising a base and power to another power,
multiply the indices. Leave the base unchanged.
Another method is to write out the full multiplication.
You try:
Expand and simplify:
Indices
5
Things to Know:
Law 5: x  a 
Describe in words:
1
xa
How it’ done:
What number does 4–2 represent?
4–2 is the reciprocal of 42.
42 = 16.
4–2 =
So 4–2 =
=
The negative index applies to the pronumeral p
only.
Express 4p–3 as a fraction.
4p–3 = 4 × p–3 = 4 ×
=
p–3 =
You try:
Simplify, leaving answer with a positive index:
Indices
6
A more complex example:
Simplify
form without fraction notation.
÷ 6m4= 24m–5 ÷ 6m4= 4m-5 -4= 4m- 9
Converting an expression from fractions to index
index laws.
Note that this answer expressed in fraction form
would be
.
You try:
Indices
7
Simplify:
Lesson 1 Homework: 3F (p. 186): # 4, 6, 8, 11 ; 3G (p. 191): # 1-3, 5, 9, 10
Indices
8
Part 2: Rational (Fractional) indices:
Things to Know:
The same laws that apply to natural number indices, apply to rational (fractional) indices
How it’s done:
Express
in root form then as a basic numeral.
is the fifth root of 32.
A base raised to the power
base.
is the mth root of that
=
So
Simplify (
)8=
(
=
Taking roots and raising to powers are inverse
operations, just like ÷ and × are inverse operations.
)8.
×8
= k4
When raising a power to a further power, multiply the
indices.
(am)n = a m × n = amn
You can also apply this to roots and powers.
=
So (
)8= (
)8
= k4
Simplify p × p × q ÷ q .
When simplifying an expression containing several
pronumerals, deal with each pronumeral separately.
The index laws apply to all types of indices:
p ×p ×q÷q

= p1 × q ÷ q

when multiplying terms with the same base, add
the indices
when dividing terms with the same base, subtract
the second index from the first index
=p×q÷q
=p×q
= pq
You try:
Indices
9
Evaluate:
Use your understanding of index laws and indices to evaluate
By writing roots in index form, you can evaluate
this expression without the need for a calculator
)–4.
(
=
)–4 = ( )–4
(
= 5–2
=
=
You Try:
Lesson 2 Homework: 3I (p. 200): # 5, 8-10
Indices
10
Part 3: Scientific notation
Things to Know:
1. To express a large number in scientific notation, write it in the form:
(NUMBER BETWEEN 1 AND 10) × 10POWER
2. To express a small number in scientific notation, write in form
(NUMBER BETWEEN 1 AND 10) × 10 –POWER
How it’s done:
4×2=8
Calculate 4 × 10–3 × 2 × 108.
10–3 × 108 = 105
5
5
66826111
0
2
You could also write both numbers as basic
numerals before calculating.
5
5
4 × 10–3=4 ×
2
=
–3
8
4 × 10 × 2 × 10 = 8 × 10
5
2 × 108 =200 000 000
You try:
Write in Scientific notation:
3. Calculate, leaving answer in scientific notation:
Indices
11
Lesson 3 Homework: 3H (p. 195): # 4-6 , 7
Indices
12
```