Classwork Exercise and Series (Mathematics- JSS 2): Indices and Standard Form

Indices

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

What are Indices?

The expression 2⁵ is defined as follows: 2⁵ = 2 x 2 x 2 x 2 x 2

We call “2” the base and “5” the index.

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).

Six rules of the Law of Indices

Rule 1: a⁰ = 1

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

An Example:

Simplify 2⁰:

2⁰ = 1

Rule 2:a^-m = 1/a^m

An Example:

Simplify 2^-2:

2^-2 = 1/2²     Using a^-m = 1/a^m

           = ¼

Rule 3: a^m x aⁿ  = a^m+n

To multiply expressions with the same base, copy the base and add the indices.

An Example:

Simplify 5 x 5³: (note: 5 = 5¹)

5¹ x 5³ = 5¹⁺³                        Using a^m x aⁿ = a^m+n

= 5⁴

= 5 x 5 x 5 x 5 = 625

Rule 4: a^m ÷ aⁿ = a^{m – n}

To divide expressions with the same base, copy the base and subtract the indices.

Example:

Simplify 5(y⁹ ÷ y⁵)

5(y⁹ ÷ y⁵) = 5(y^9-5):                       Using a^m ÷ aⁿ = a^m-n

= 5y⁴

Rule 5: (a^m)ⁿ = a^mn

To raise an expression to the nth index, copy the base and multiply the indices.

Example

Simplify (y²)⁶:

(y²)⁶ = y^{2 x 6}                  Using (a^m)ⁿ =a^mn

= y¹²

Rule 6: a^m/n  = ⁿ√a^m  = (ⁿ√a)^m

Example:

Simplify 125^2/3 = (³√125)^m               Using a^m/n = ⁿ√a^m  = (ⁿ√a)^m

= 5²                                           Recognized cube root of 125 is 5

= 25

Standard Form

Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative.

The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Example

Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10¹³

It’s 10¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19

Example

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10^-6

It’s 10^-6 because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.

Manipulation in Standard Form

This is best explained with an example:

Example

The number p written in standard form is 8 × 10⁵

The number q written in standard form is 5 × 10^-2

Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:

8 × 5 × 10⁵ × 10^-2

= 40 × 10³ (Remember 10⁵ × 10^-2 = 10³)

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.

= 4 × 10⁴

Calculate p ÷ q.

Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits. (8 ÷ 5) × (10⁵ ÷ 10^-2) = 1.6 × 10⁷

Express the following fractions in standard form

a. 0.000 07

b. 0.000 000 022

c. 0.075

Rounding off numbers

You have learnt how to round off numbers to the nearest thousand, hundred, tens, etc.

Remember that the digits 1, 2, 3, 4 are rounded down and the digits 5, 6, 7, 8, 9 are rounded up.

Round off the following to the nearest

i. thousand ii. hundred iii. ten

a. 12 835

b. 46 926

c. 28 006

Significant figures

Significant figures begin from the first non-zero digit at the left of a number. As before, the digits 5, 6, 7, 8, 9 are rounded up and 1, 2, 3, 4 are rounded down. Digits should be written with their correct place value.

Read the following examples carefully.

a. 546.53 = 500 to 1 significant figure (s.f.)

543.52 = 550 to 2 s.f.

543.52 = 547 to 3 s.f.

546.52 = 546.5 to 4 s.f.

b. 8.0296 = 8 to 1 s.f.

8.0296 = 8.0 to 2 s.f.

In this case the zero must be given after the decimal point. It is important.

8.0296 = 8.03 to 3 s.f.

8.0296 = 8.030 to 4 s.f.

Notice that the fourth digit is zero. It is significant and must be written

Decimal Places

Decimal places are counted from the decimal point. Zeros after the point are significant and are also counted. Digits are rounded up or down as before. Place value must be kept.

Read the following examples carefully.

a. 14.902 8= 14.9 to 1 decimal place (d.p.)

b. 14.902 8 = 14.90 to 2 d.p.

3. 14.902 8 = 14.903 to 3 d.p.

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