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3 Indices and Standard Form
3.1 Index Notation
Here we revise the use of index notation. You will already be familiar with the notation for squares and cubes
a a a a a a a
2 3
= ×
= × ×
, and
this is generalised by defining:
a n^ = a 1 × 4 4 a (^) 2 × ... 44 × 3 a n of these
Example 1
Calculate the value of: (a) 5 2 (b) 2 5 (c) 33 (d) 10 4
Solution
(a) 5 2 = 5 × 5 = 25
(b) 2 5 = 2 × 2 × 2 × 2 × 2 = 32
(c) 33 = 3 × 3 × 3 = 27
(d) 10 4 = 10 × 10 × 10 × 10 = 10 000
Example 2
Copy each of the following statements and fill in the missing number or numbers:
(a) 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2
(b) 9 = 3
(c) 1000 = 10
(d) 53 = × ×
Solution
(d) Express your answer to (c) in index notation. Exercises
- (a) 2 7 = 2 × 2 × 2 × 2 × 2 × 2 ×
- (b) 9 = 3 × 3 =
- (c) 1000 = 10 × 10 × 10 =
- (d) 53 = 5 × 5 ×
- Example
- (a) Determine
- (b) Determine
- (c) Determine 2 5 ÷
- (a) 2 5 = Solution
- (b) 2 3 =
- (c) 2 5 ÷ 23 = 32 ÷ - =
- (d) 4 =
- (a) 2 3 (b) 10 2 (c) 1. Calculate:
- (d) 10 3 (e) 9 2 (f)
- (g) 2 4 (h) 3 4 (i)
- (a) 10 × 10 × 10 × 10 × 10 = 2. Copy each of the following statements and fill in the missing numbers:
- (b) 3 × 3 × 3 × 3 =
- Calculate: (a) (^) ( 3 + (^2) )^4 (b) (^) ( 3 − (^2) )^4
(c) (^) ( 7 − (^4) )^3 (d) (^) ( 7 + (^4) )^3
- Writing your answers in index form, calculate: (a) 10 2 × 103 (b) 2 3 × 27 (c) 3 4 ÷ 32 (d) 2 5 ÷ 22 (e) 10 6 ÷ 102 (f) 5 4 ÷ 52
- (a) Without using a calculator, write down the values of k and m. 64 = 8 2 = 4 k^ = 2 m
(b) Complete the following: 2 15 = 32 768 2 14 = (KS3/99Ma/Tier 5-7/P1)
3.2 Laws of Indices
There are three rules that should be used when working with indices:
When m and n are positive integers,
- a m^ × a n^ = am^ + n
- a m^ ÷ a n^ = am^ −^ n or a a
a
m n = m^ −^ n ( m ≥ n )
- (^) ( a m^ ) n^ = am^ × n
These three results are logical consequences of the definition of a n^ , but really need a formal proof. You can 'verify' them with particular examples as below, but this is not a proof: 2 7 × 23 = ( 2 × 2 × 2 × 2 × 2 × 2 × 2 ) × ( 2 × 2 × 2 ) = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 10 (here m = 7 , n = 3 and m + n = 10 )
or,
2 7 ÷ 23 = 2 2 2 2 2 2 2 2 2 2
× × × × × ×
× ×
= 2 × 2 × 2 × 2
= 2 4 (again m = 7 , n = 3 and m − n = 4 )
Also, (^) ( (^2 7) ) 3 = 2 7 × 2 7 × 27
= 2 21 (using rule 1) (again m = 7 , n = 3 and m × n = 21 )
The proof of the first rule is given below:
Proof
a m^ × an = a 1 × 4 4 a (^) 2 × ... 44 × 3 a × a 1 × 4 4 a (^) 2 × ... 44 × 3 a m of these n of these
= a 1 × 4444 a × ... × 4 a 2 × 44444 a × a ...× 3 a (^ m^ + n ) of these
= a m^ + n
The second and third rules can be shown to be true for all positive integers m and n in a similar way.
We can see an important result using rule 2:
x x
x x
n n = n^ −^ n =^0
but x x
n n =^1 ,^ so x^0 = 1
This is true for any non-zero value of x, so, for example, 3 0 = 1 , 27 0 = 1 and
10010 = 1.
Exercises
- Copy each of the following statements and fill in the missing numbers:
(a) 2 3 × 2 7 = 2 (b) 3 6 × 3 5 = 3
(c) 3 7 ÷ 3 4 = 3 (d) 8 3 × 8 4 = 8
(e) (^) ( (^3 2) ) 5 = 3 (f) (^) ( (^2 3) ) 6 = 2
(g) 3 3
6 2 =^ (h)^
7 2 =
- Copy each of the following statements and fill in the missing numbers:
(a) a^3 × a^2 = a (b) b^7 ÷ b^2 = b
(c) (^) ( b^2 ) 5 = b (d) b^^6 ×^ b^^4 = b
(e) (^) ( z^3 ) 9 = z (f) q q
q
16 7 =
- Explain why 9 4 = 38.
- Calculate:
(a) 3 0 + 40 (b) 6 0 × 70 (c) 8 0 − 30 (d) 6 0 + 2 0 − 40
- Copy each of the following statements and fill in the missing numbers:
(a) 3 6 × 3 = 317 (b) 4 6 × 4 = 411
(c) a a
6 = a^4 (d) (^) ( z (^) ) 6 = z^18
(e) (^) ( a^19 ) = a 9 5 (f) p^16 ÷ p = p^7
(g) (^) ( p (^) ) 8 = p^40 (h) q^13 ÷ q = q
- Calculate:
(a) 2 2
3 2
4 3
−^0
(c) 5 5
4 2
2
7 5
9 − 7
(e) 10 10
8 5
6 − 3 (f)
17 14
13 − 11
- Fill in the missing numbers in each of the following expressions:
(a) 8 2 = 2 (b) 81 3 = 9 = 3
(c) 25 6 = 5 (d) 4 7 = 2
(e) 125 4 = 5 (f) 1000 6 = 10
(g) 81 = 4 (h) 256 = 4 =^8
- Fill in the missing numbers in each of the following expressions:
(a) 8 × 4 = 2 × 2 (b) 25 × 625 = 5 × 5
= 2 = 5
(c) 243 9
(d) 128 16
- Is each of the following statements true or false?
(a) 3 2 × 2 2 = 64 (b) 5 4 × 2 3 = 107
(c) 6 2
8 8 = 8 (d) 10 5
8 6
=^2
and, in general, a −^ n = 1 a n
for positive integer values of n. The three rules at the start of section 3.2 can now be used for any integers m and n, not just for positive values.
Example 1
Calculate, leaving your answers as fractions: (a) 3 −^2 (b) 2 −^1 − 4 −^1 (c) 5 −^3
Solution
(a) 3 −^2 = 1 3 2 = 1 9
(b) 2 −^1 − 4 −^1 = 1 2
(c) 5 −^3 = 1 5 3 = 1 125
Example 2
Simplify:
(a) 6 6
7 9 (b)^6
(^4) × − (^3) (c) (^) ( (^10 2) ) −^3
Solution
(a) 6 6
7 9 =^6
7 − 9
= 6 −^2 = 1
(b) 6 4 × 6 −^3 = 6 4 +^ (−^3 ) = 6 4 −^3 = 6 1 = 6
(c) (^) ( (^10 2) ) −^3 = 10 −^6
= 1 10 6
= 1 1000 000
Exercises
- Write the following numbers as fractions without using any indices :
(a) 4 −^1 (b) 2 −^3 (c) 10 −^3 (d) 7 −^2 (e) 4 −^3 (f) 6 −^2
- Copy the following expressions and fill in the missing numbers:
(a) 1 49
= = 7 (b) 1 100
(c) 1 81
= = 9 (d) 1 16
(e) 1 10 000 000
= = 10 (f) 1 1024
- Calculate:
(a) 4 −^1 + 3 −^1 (b) 6 −^1 + 2 −^1 (c) 5 −^1 − 10 −^1 (d) 10 −^2 − 10 −^3 (e) 4 −^1 − 10 −^1 (f) 6 −^1 + 7 −^1
- Simplify the following expressions giving your answers in the form of a number to a power: (a) 4 7 × 4 −^6 (b) 5 7 × 5 −^3
(c) 7 7
4 − 6 (d)^3 ( 2 ) −^4
(e) (^) ( 6 −^2 ) −^3 (f) 8 4 × 8 −^9
(g) 7 7
2 − 2 (h)^
9 − 9
- Copy the following expressions and fill in the missing numbers:
(a) 1 8
= 2 (b) 1 25
(c) 1 81
= 9 (d) 1 10 000
- If a b b c
= 3 and =^12 , express a as a power of c, without having any fractions in your final answer.
3.4 Standard Form
Standard form is a convenient way of writing very large or very small numbers. It is used on a scientific calculator when a number is too large or too small to be displayed on the screen. Before using standard form, we revise multiplying and dividing by powers of 10.
Example 1
Calculate: (a) 3 × 10 4 (b) 3 27. × 10 3 (c) 3 ÷ 10 2 (d) 4 32. ÷ 10 4
Solution
(a) (^3) × 10 4 = (^3) ×10 000 = 30 000
(b) 3 27. × 10 3 = 3 27. × 1000 = 3270
(c) 3 ÷ 10 2 = 3 100 = 0.
(d) 4 32. ÷ 10 4 = 4 32 10 000
These examples lead to the approach used for standard form, which is a reversal of the approach used in Example 1.
In standard form , numbers are written as a × 10 n where 1 ≤ a < 10 and n is an integer.
Example 2
Write the following numbers in standard form: (a) 5720 (b) 7. (c) 473 000 (d) 6 000 000 (e) 0.09 (f) 0.
Solution
(a) 5720 = 5 72. × 1000 = 5 72. × 10 3
(b) 7.4 = 7 4. × 1 = 7 4. × 10 0
(c) 473 000 = 4 73. ×100 000
= 4 73. × 10 5
(d) 6 000 000 = 6 ×1000 000 = 6 × 10 6
(e) 0.09 = 9 100 = 9 ÷ 10 2 = 9 × 10 −^2
which will appear on your display like this:
Some calculators also display the ' × 10 ' part of the number, but not all do. You need to find out what your calculator displays. Remember, you must always write the ' × 10 ' part when you are asked to give an answer in standard form.
Exercises
- Calculate: (a) 6 21. × 1000 (b) 8 × 10 3 (c) 4 2. × 10 2
(d) 3 ÷ 1000 (e) 6 ÷ 10 2 (f) 3 2. ÷ 10 3
(g) 6 × 10 −^3 (h) 9 2. × 10 −^1 (i) 3 6. × 10 −^2
- Write each of the following numbers in standard form: (a) 200 (b) 8000
(c) 9 000 000 (d) 62 000
(e) 840 000 (f) 12 000 000 000 (g) 61 800 000 000 (h) 3 240 000
- Convert each of the following numbers from standard form to the normal decimal notation: (a) 3 × 10 4 (b) 3 6. × 10 4 (c) 8 2. × 10 3 (d) 3 1. × 10 2 (e) 1 6. × 10 4 (f) 1 72. × 10 5 (g) 6 83. × 10 4 (h) 1 25. × 10 6 (i) 9 17. × 10 3
- Write each of the following numbers in standard form: (a) 0.0004 (b) 0. (c) 0.142 (d) 0. (e) 0.00199 (f) 0. (g) 0.0000097 (h) 0.
- Convert the following numbers from standard form to the normal decimal format: (a) 6 × 10 −^2 (b) 7 × 10 −^1 (c) 1 8. × 10 −^3 (d) 4 × 10 −^3 (e) 6 2. × 10 −^3 (f) 9 81. × 10 −^4 (g) 6 67. × 10 −^1 (h) 3 86. × 10 −^5 (i) 9 27. × 10 −^7
- Without using a calculator, determine:
(a) (^) ( 4 × (^10 4) ) × (^) ( 2 × (^105) ) (b) (^) ( 2 × (^10 6) ) × (^) ( 3 × (^105) )
(c) (^) ( 6 × (^10 4) ) × (^) ( 8 × 10 −^9 ) (d) (^) ( 3 × 10 −^8 ) × (^) ( 7 × 10 −^4 )
(e) (^) (6 1. × (^10 6) ) × (^) ( 2 × 10 −^5 ) (f) (^) (3 2. × 10 −^5 ) × (^) ( 4 × 10 −^9 )
- Without using a calculator, determine:
(a) (^) ( 9 × (^10 7) ) ÷ (^) ( 3 × (^104) ) (b) (^) ( 8 × (^10 5) ) ÷ (^) ( 2 × 10 −^2 )
(c) (^) ( 6 × 10 −^2 ) ÷ (^) ( 2 × 10 −^3 ) (d) (^) ( 6 × (^10 4) ) ÷ (^) ( 3 × 10 −^6 )
(e) (^) ( 4 8. × (^10 12) ) ÷ (^) (1 2. × (^103) ) (f) (^) (3 6. × (^10 8) ) ÷ (^) ( 9 × (^103) )
- Without a calculator, determine the following, giving your answers in both normal and standard form:: (a) (^) ( 6 × (^10 5) ) + (^) ( 3 × (^106) ) (b) (^) ( 6 × (^10 2) ) + (^) ( 9 × (^103) )
(c) 6 × 10 5 − 1 × 104 (d) 8 × 10 −^2 + 9 × 10 −^3 (e) 6 × 10 −^4 + 8 × 10 −^3 (f) 6 × 10 −^4 − 3 × 10 −^5
- Use a calculator to determine:
(a) (^) (3 4. × (^10 6) ) × (^) (2 1. × (^104) ) (b) (^) ( 6 × (^10 21) ) × (^) (8 2. × 10 −^11 )
(c) (^) (3 6. × (^10 5) ) × (^) (4 5. × (^107) ) (d) (^) (8 2. × (^10 11) ) ÷ (^) ( 4 × 10 −^8 )
(e) (^) (1 92. × (^10 6) ) × (^) (3 2. × 10 −^11 ) (f) (^) (6 2. × (^10 14) )^3
- (a) Which of these statements is true? (i) 4 × 10 3 is a larger number than 4 3. (ii) 4 × 10 3 is the same size as 4 3. (iii) 4 × 10 3 is a smaller number than 4 3. Explain your answer.
(b) One of the numbers below has the same value as 3 6. × 10 4. Write down the number. 36 3 36 4 (3 6. × 10 ) 4 0 36. × 10 3 0 36. × 105
(c) One of the numbers below has the same value as 2 5. × 10 −^3. Write down the number. 25 × 10 −^4 2 5. × 10 3 − 2 5. × 10 3 0 00025. 2500
(d) (^) ( 2 × (^10 2) ) × (^) ( 2 × (^102) ) can be written more simply as 4 × 10 4. Write the following values as simply as possible:
(i) (^) ( 3 × (^10 2) ) × (^) ( 2 × 10 −^2 )
(ii) 6 10 2 10
8 4
×
×
(KS3/98/Ma/Tier 6-8/P1)
3.5 Fractional Indices
Indices that are fractions are used to represent square roots, cube roots and other roots of numbers.
a
1 (^2) = a for example, 9
1 (^2) = 3
a
1 (^3) = 3 a for example, 8
1 (^3) = 2
a
1 (^4) = 4 a for example, 625
1 (^4) = 5
a n
1 = na
Example 1
Calculate:
(a) 81
1 (^2) (b) 1000
1 (^3) (c) 4
1 − 2
Solution
(a) 81
1 (^2) = 81 = 9
(b) 1000
1 (^3) = 31000 = 10
(c) 4
1 − (^2) = 1
4
1 2
= 1 4
= 1 2
Exercises
- Calculate:
(a) 49
1 (^2) (b) 64
1 (^2) (c) 16
1 2
(d) 81
1 − (^2) (e) 100 1 − (^2) (f) 25 1 − 2
(g) 9
1 (^2) (h) 36
1 − (^2) (i) 144 1 2
- Calculate:
(a) 8
1 (^3) (b) 8
1 − (^3) (c) 125 1 3
(d) 64
1 − (^3) (e) 216 1 (^3) (f) 1000 000
1 − 3
- Calculate:
(a) 32
1 (^5) (b) 64
1 − (^2) (c) 10 000 1 4
(d) 81
1 − (^4) (e) 625 1 (^4) (f) 100 000
1 − 5
