Module 1
Indices / Year 10A
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
Homework: 3I (p. 200): # 5, 8-10
Lesson 3: Scientific Notation
Homework: 3H (p. 195): # 4-6 /
Materials
Essential mathematics3.6 - 3.9
Time Required
3 periodsTest date:______l
Part 1: Indices Laws:
Things to Know:
Law 1: / Describe in words:How it’s done:
Simplify 42 × 45 giving your answer in index form.42 × 45 / = / 42 + 5
= 47Top of Form / When multiplying terms with the same base, you add the indices.
The bases do not change.
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.
You try:
Things to Know:
Law 2: / Describe in words:How it’s done:
Simplify 612 ÷ 63.612 ÷ 63 = 6 12 – 3
= 69
/ 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.
Simplify 15p10 ÷ 3p.
15p10 ÷ 3p / = / 5p10 ÷ p1 / = 5p10 – 1 = 5p9
/ Remember that when the index is 1 it can be omitted.
You try:
Simplify:
You try:
Simplify:
Things to Know:
Law 3: / 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
- addition & subtraction
(Remember the debate about 00 which might be undefined or might be 1 depending on how you look at it.)
You try:
Simplify:
Things to Know:
Law 4: / Describe in words:How it’s done:
Write (73)2 as an equivalent expression in simple index form.(73)2= 73×2= 76
Expand (h5)4.
(h5)4 = h5 × 4 =h9
/ 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.
Expand (3y7)3.
(3y7)3 = 33 × y7× 3 =27y21
/ 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:
Things to Know:
Law 5: / Describe in words:How it’ done:
What number does 4–2 represent?4–2 = =
Express 4p–3 as a fraction.
4p–3 = 4 × p–3 = 4 × =
/ 4–2 is the reciprocal of 42.
42 = 16. So 4–2 =
The negative index applies to the pronumeralp only.
p–3 =
You try:
Simplify, leaving answer with a positive index:
A more complex example:
Simplify ÷ 6m4. Express your answer in index form without fraction notation.÷ 6m4= 24m–5 ÷ 6m4= 4m-5 -4= 4m- 9 / Converting an expression from fractions to index form can help you simplify calculations using index laws.
Note that this answer expressed in fraction form would be .
You try:
Simplify:
Lesson 1 Homework: 3F (p. 186): # 4, 6, 8, 11 ;3G (p. 191): # 1-3, 5, 9, 10
Part 2: Rational (Fractional) indices:
Things to Know:
The same laws that apply to natural number indices, apply to rational (fractional) indicesHow it’s done:
Express in root form then as a basic numeral.is the fifth root of 32.
So =
/ A base raised to the power is the mth root of that base.
=
Taking roots and raising to powers are inverse operations, just like ÷ and × are inverse operations.
Simplify ()8.
()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
Simplifyp × p × q ÷ q.
p × p × q ÷ q
= p1 × q ÷ q
= p × q ÷ q
= p × q
= pq / When simplifying an expression containing several pronumerals, deal with each pronumeral separately.
The index laws apply to all types of indices:
- 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
You try:
Evaluate:
Use your understanding of index laws and indices to evaluate ()–4.=
()–4 / = ()–4
= 5–2
=
=
/ By writing roots in index form, you can evaluate this expression without the need for a calculator
You Try:
Lesson 2 Homework: 3I (p. 200): # 5, 8-10
Part 3: Scientific notation
Things to Know:
- To express a large number in scientific notation, write it in the form:
(NUMBER BETWEEN 1 AND 10) × 10POWER
- To express a small number in scientific notation, write in form
How it’s done:
Calculate 4 × 10–3 × 2 × 108.Express your answer in scientific notation.Top of Form
Bottom of Form
Top of Form
4 × 10–3 × 2 × 108 = 8 × 105
Bottom of Form / 4 × 2 = 8
10–3 × 108 = 105
You could also write both numbers as basic numerals before calculating.
4 × 10–3 / = / 4 ×
= /
2 × 108 / = / 200 000 000
You try:
Write in Scientific notation:
- Calculate, leaving answer in scientific notation:
Lesson 3 Homework: 3H (p. 195): # 4-6 , 7