WEEK 5:                                                                        

DATE……………………….

TOPIC:

Standard form and Indices

Content:

• Revision of standard form
• Introduce indices and examples
• Laws of indices
• Application of indices, simple indicial equation

SUB-TOPIC    1: Revision    of standard    form

Example 1; Express the following in standard form

(a) 5.37 (b) 53.7 (c) 537 (d) 35.65 (e) 7500 (f) 1403420

Solution:

(a) 5.37 = 5.37 x 1

= 5.37 x

(b) 53.7 = 5.37 x

(c) 537 = 5.37 x 100

= 5.37 x 10 x 10

= 5.37 x

(d) 35.65 = 3.565 x 10

= 3.565 x

(e) 7500 = 7.5 x 1000

= 7.5 x

(f) 1403420 = 1.403420 x 1000000

= 1.403420 x

Example 2; Express the following in standard form

(a) 0.037 (b) 0.00065 (c) 0.0058 (d) 0.61

Solution:

(a) 3.7 x 10^{-2}

(b) 6.5 x 10^{-4}

(c) 5.8 x 10^{-3}

(d) 6.1 x 10^{-1}

Method 1:

(a) 0.037 = 3.7 x 0.01

= 3.7 x

(b) 0.00065 = 6.5 x 0.0001

= 6.5 x

(c) 0.0058 = 5.8 x 0.001

= 5.8 x

(d) 0.61 = 6.1 x 0.1

= 6.1 x

Express the following numbers in stand- ard form:

1

(a) 6000

(b) 7000

(e) 80 000

(d) 52

(e) 46

(D3

(g) 520

(h) 6400

(1) 2

() 4000

(k) 152

(1) 64 000 000

Here are the numbers expressed in standard form:

(a) 6000 = 6 x 10^3

(b) 7000 = 7 x 10^3

(c) 80,000 = 8 x 10^4

(d) 52 = 5.2 x 10^1

(e) 46 = 4.6 x 10^1

(f) 520 = 5.2 x 10^2

(g) 6400 = 6.4 x 10^3

(h) 2 = 2 x 10^0 (Note: Any non-zero number to the power of 0 is 1)

(i) 4000 = 4 x 10^3

(j) 152 = 1.52 x 10^2

(k) 64,000,000 = 6.4 x 10^7

EVALUATION:

Express the following in standard form

1. (a) 86000    (b) 4730     (c) 307    (d) 1903000
2. (a) 0.075     (b) 0.00059     (c) 0.22    (d) 0.0000036

SUB-TOPIC    2: Introduction of indices and examples

The law of indices, or exponentiation law, is a mathematical rule that defines how to calculate with numbers that are raised to a power. In mathematics, an index (or power) indicates how many times a number is multiplied by itself. For example, the expression “23” simply means “2 multiplied by two, three times”. The number 3 is called the index or power.

The law of indices states that when two numbers with the same base are raised to a power, the powers can be added. So, for example,

25 = 2 × 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 2

25 + 24 = 27

This law can be extended to more than two numbers with the same base. For example,

23 = 2 × 2 × 2

22 = 2 × 2

21 = 2

23 + 22 + 21 = 27

The law of indices can also be used to simplify expressions that involve powers of the same base. For example, the expression “23 24” can be simplified using the law of indices:

23 × 24 = 2(3+4) = 27

This law is also known as the law of exponents or the exponentiation law.

Laws Of Indices

The following are the laws governing mathematical operations involving index numbers. These laws are true for all values of m, n and x ≠ 0.

• x ^m× x ⁿ =         x ^{m+ n}
• x ^m _÷ x ⁿ =         x ^m / x ⁿ = x ^{m –n}
• x ^–n =         ¹/ x ⁿ

x 0 = 1

(xy) n = x n y n

(x/y) n = x n / y n

x –m = 1 / x m

The zero power rule: Anything to the power of zero is equal to one

The negative exponent rule: Anything to the power of a negative is equal to one over that number

The product rule: When two powers with the same base are multiplied, add their exponents

The power of a power rule (or exponential growth): When a power is raised to a power, multiply the exponents

The quotient of two powers with the same base: When a base is both divided and raised to a power, subtract

Indices is the plural of the word index. An index is the power of a given number. Numbers are sometimes expressed in index form e.g. 8 can be expressed as 2³ in index form, 81 can be

expressed as 3⁴ in index form, ¹/₁₂₅ can be expressed as ¹/ ³ or 5^-3 in index form etc.

A number when expressed in index form must have a base and a power, e.g. when 9 is expressed in index form, we have 3²_. In this case, 3 is the base and 2 is the index or power

.When 625 is expressed in the index form, we have 5⁴ Here, 5 is the base and 4 is the index or power.

In general, index numbers are written in the form x^m where x is the base and m is the index. It should be noted that mathematical operations (ie +, _ ,× and ÷) involving these types of numbers does not follow the conventional way. There are laws that govern its mathematical operations.        Hence, this topic indices deals with mathematical operations involving index numbers.

SUB-TOPIC    3: Laws Of Indices

The following are the laws governing mathematical operations involving index numbers. These laws are true for all values of m, n and x ≠ 0.

• x ^m× x ⁿ =          x ^{m+ n}
• x ^m _÷ x ⁿ =          x ^m / x ⁿ = x ^{m –n}
• x ^–n =          ¹/ x ⁿ

(4) x ^o               =          1

(5) (x ^m)ⁿ          =          x ^m.n

(6) x^1/n             =

(7) x ^m/n            =          (x ^m) ^1/n

=

SUB-TOPIC    4: Applications        Of The Laws

The following are some examples of how to apply the laws of indices stated above.

LAW 1: x ^m X x ⁿ = x ^{m + n}

Example 1: Simplify the following

{i} 5² x 5⁷                    {ii} 3² x 3⁴ x 3³ x 3

{iii} 2³a^-2 b x 2a⁵b³       {iv} x^3/4 X x^5/8

Solution:

(i) 5² x 5⁷ = 5²⁺⁷

= 5⁹

(ii) 3² x 3⁴ x 3³ x 3 = 3^2+4+3+1

= 310 (Since 3 = 31)

• 2³a^-2bx 2a⁵b³= 2³ x 2¹ xa^-2 xa⁵ xb¹ xb³

= 23+1 a-2+5 b1+3

= 2⁴a³b⁴

= 16a³b⁴

(iv) x3/4 X x5/8    = x(6+5)/8

= x11/8

LAW 2: x ^m ÷ x ⁿ = x ^m-n

Example 2:

simplify the following (i) 3⁷ ÷ 3⁴

• 21a⁴ b³ 7ab²

• 9a⁵ b33a²b^-2

Solution^:

(i)3⁷ ÷3⁴           =3^{7 – 4}

=3³

= 27

• 21a⁴b³

7ab2                   = 3a4 – 1b3 – 2

= 3a³b

(iii) 9a⁵b³ ÷ 3a²b^-2 = 3a^{5 – 2} b^{3 – (-2)}

= 3a3 b3 + 2

= 3a³b⁵

EVALUATION

Simplify the following

(i) 73 x 76

• 25 x 23 x 22
• 34a-2b x 3a4b2
• a1/2 x a1/4

(v) 5⁴¸ 5⁷

• 15a³b⁵ 5ab²
• 8a⁶b²¸ 4ab^-2

LAW 3: x ^–n = ¹/xⁿ

Example 3:

Simplify the following

(i) 2^-4                (ii)        6a^-2b³ 3a3 b-5

(iii) (²⁷/₈ )^-2/3

(iv) 24x⁴y⁶        (v) _1_ 8x9y3   3-2

Solution:

(i)         2^-4 = _1_ 24

= _1_

16

• 6a^-2b³ = 6b³b⁵ 3a³b^-5 3a³a²

= 2b3+5

a3+2

= 2b⁸

a5

(iii)     27 ^-2/3       8 ^2/3

8     =     27

=

2/3

3 3×2/3

= 2²

32

= 4

9

(iv)     24x⁴y⁶ = 3x⁴ . x^{– 9}. y⁶ . y^{– 3}

8x⁹y³

= 3×4-9 y6-3

= 3x^-5y³

= 3y³

x5

(v)        _1_

3-2              = 32

= 9

LAW 4:            x⁰ = 1

Example 4:

Simplify the following

(i)         (20ab⁷ x 15a⁶bc⁵)⁰

(ii)        (17x⁴y²)⁰ + 1

• 3² x 3^-3 x 3

Solution:

(i)         (20ab⁷ x 15a⁶bc⁵)⁰ = 1 (ii)    (17x⁴y²)⁰ + 1    = 1 + 1

= 2

(iii)       3² x 3^-3 x 3       = 3² x 3^-3 x 3¹

= 32-3+1

= 3⁰

= 1

LAW 5:

(xm)n        =          xm.n

Example 5:

Simplify the following (i)      (ab²)³ x (2a⁴b)²

(ii)        5a³b² x (2ab)^-2

(iii)       (3x²)³ ÷ 9x^-3

Solution:

(i)         (ab²)³ x (2a⁴b)²

= a3 b2x3 X 22 X a4x2 b2

= a³b⁶ X 4a⁸b²

= 4a3+8 b6+2

= 4a¹¹b⁸

• 5a3b2 x (2ab)-2

=          5a3b2 x 2-2a-2b-2

=          5 x 2^-2a³ x a^-2 x b² x b^-2

=          5 x 1 a3-2 b2-2

22

=          5 ab⁰

4

=     5a 4

(iii)       (3x²)³ ÷ 9x^-3

=          33x2x3 ÷ 9x-3

=           27x⁶ ÷ 9x^-3

=          3×6 – (-3)

=          3×6 + 3

=          3x⁹

EVALUATION

Simplify the following

(i) 3^-5          (ii) 2a^-3b² 3a2b-4

• 15a³b⁵ (iv) 30x³y⁵ 5ab² 6x⁷y²

(v)      3^-4 2

(vi) (300ab²)⁰

• (27xy4)0 + 1
• 2³ x 2^-4 x 2
• (x³y)² x (2xy²)⁵
• 2(ab²)³ x a²b
• 3a²b x (2ab)^-3

LAW 6: x^1/n    =

Example 6

Simplify the following. (i.)

(ii.)

(iii.)      (0.027)^1/3

Solution:

(i.)

= (33)1/3

= 33 x 1/3

(ii).

=   3

1/3

=

=

= 4 2 x ½

5 2 x ½

= 4

5

(iii)       (0.027)^1/3 =        0.027 ^1/3

1

=        _27_ ^1/3

1000

=        33 1/3

10³

=         33 x 1/3

103 x 1/3

=           _3_

10

LAW 7:    x^m/n = (x^m)^1/n

=

Simplify the following.

• 3 a⁸ x 2a⁶ x 4a

• ³

8y-6

Solution:

(i)

= 2x4x a8 +6-2     1/3

=       23a12 1/3

(ii)       ³ 8y^-6      = 8y^{-6 1/3}

=      8a121/3

= 23×1/3a12x1/3

= 2a⁴

=    23y-6 1/3

= 23×1/3xy-6×1/3

= 2y^-2

= 2/ y2

EVALUATION

Simplify the following

(i) ₃ 8²

Ö 27

(ii) 4 51/16-3

Ö

(iii)     3³/ ^-2/3

(iv)          (v)

(vi)

(vii)     (¹/₈₁)^-1/4

(viii)   (ix) 0.09^1/2

SUB-TOPIC    5: simple   indicial    equation

We shall consider the application of the laws of indices in solving index equations

Example

(i)          2x²      = 50

(ii)           3^x-1   = 81

(iii)           8^x     = 64

Solution:

(i)         2x²       =          50

x²         =          50

2

x²         =          25

x²         =          5²

∴   x          =          5

(ii)  3^{x – 1} = 81 3x – 1 = 34 x – 1 = 4

x    =     4 + 1

∴ x     =     5

• 8^x =          64

23x               =          26

3x         =          6

x          =          ⁶/₃

∴ x         =          2

EVALUATION

Solve the following index equation

1.  3^2x = 27
2. 9^x = 81
3. 5a³ = 40
4. a-1/2 = 5
5. 4^x = 32
6. x ^-2 = 4
7. 4^{x + 1} = 64
8. 4^x x 8 = 64
9. 3^2x-1 = 27
10. 27 x 3^x = 81
11. 2x^1/3 = 16
12. 3x² = 27

GENERAL EVALUATION

• Given that3x91+x=27-x, find x
• Given that32x= 27-9x, find x.
• Given that4a3 = 40-5a, find a.
• Given that16v = 4096 – 6v4, find v.
• Given that4x = 32 – 3×2, find x.
• Given that12y = 144 – 4y

SSCE, June 1994.

• If 4x = 2^½ x 8, find x [WAEC]
•  If 8x/2 = 23/8 x 43/4, find x [WAEC]
• If 2(a + b) = a(a + 4b), find the value of (ab)
• If 3(2x – 5) = 4(x + 1), find the value of x
• In ∆ABC, if a:b:c= 2:3:4 and B= 60°, find A and C
• In trapezium PQRS, if PS|| QR, PQ= 9cm, QR= 12cm and SR= 8cm, find the area of ∆PSR.
• Given that 4^x = 32, find the value of 1985[WAEC]

• Solve the equation 125^x+3 = 5, find

1981[WAEC]

• Given that 8 x 4^x-2 = 64, find
• Given that 125^2x+1 = 625 x 25^-x, find

(8) If 3^m x 27^(2m-1) = 81, find m.

[WAEC]

• Find the value of x in 8^-1 x 2^(2x+1) = 64.      [WAEC].

• Find the value of x, given that 7 x 49(x+2) =

The law of indices is a mathematical rule that governs the behavior of numbers when they are raised to a power. It states that when a number is raised to a power, the resulting value is equal to the product of the number and all of the preceding integers up to and including the exponent. In other words, the law of indices says that a number raised to a power is equal to the number multiplied by itself as many times as the exponent dictates.

For example, according to the law of indices, 8 raised to the 3rd power is equal to 8x8x8, or 512. This is because 8 multiplied by itself 3 times equals 512. Similarly, 9 raised to the 4th power is equal to 9x9x9x9, or 6561.

The law of indices can be used to simplify expressions that involve powers. For instance, if you wanted to calculate the value of 8 raised to the 5th power, you could first use the law of indices to rewrite the expression as 8x8x8x8x8. This is much easier to calculate than 8 raised to the 5th power, which would be equal to 32768.

Here are five examples of the law of indices in action:

1. 3 raised to the 2nd power is equal to 3×3, or 9.

2. 4 raised to the 3rd power is equal to 4x4x4, or 64.

3. 5 raised to the 4th power is equal to 5x5x5x5, or 625.

4. 6 raised to the 5th power is equal to 6x6x6x6x6, or 7776.

5. 7 raised to the 6th power is equal to 7x7x7x7x7x7, or 117649.

Presentation

The topic is presented step by step

Step 1:

The subject teacher revises the previous topics

Step 2.

He or she introduces the new topic.

Step 3:

The subject teacher allows the pupils to give their own examples and he corrects them when the needs arise