Laws of Exponents & Use of Exponents to Express Small Numbers in Standard Form – Exponents and Powers | Class 8 Maths

It is known that numbers can be expressed as xⁿ where ‘x’ is called “Base” and ‘n’ is called as “Exponent”. In simple words, we can say that the Significance of the exponent is that it tells the number of times we need to multiply our base. There are certain Laws of exponents which will make the calculation easier and faster. Let’s see Laws along with examples, in all the examples we have taken x as 5 for better understanding, x can be any number.    

Laws	Examples
x0 = 1	50  = 1
x1 = x	51  = 5
x-1 = 1 / x	5-1  = 1 / 5
xa xb = x(a + b)	54 53 = 5(4 + 3) = 57
xa / xb = x(a – b)	56 / 52  = 5(6 – 2) = 54
(xa)b = x(a * b)	(53)4 = 5(3 * 4) = 512
(xy)a = xaya	(5 * 6)2 = 52 * 62
(x/y)a = xa/ya	(5/6)3 = 53/6 3
x-a  = 1/xa	5-4 = 1/54

Let’s Discuss each of the laws in more detail

Law 1

If we have any number as base and exponent of that base is 0 then answer will be 1 .

For Example:  

2⁰ = 1 

 3⁰ =1

12⁰ = 1

Law 2

If we have any number as base and exponent of that base is 1 the answer is base itself.

For Example:

7¹ = 7 

21¹ = 21

15¹ = 15

Law 3 

If we have any number as base and exponent of that base is -1 then  answer will be reciprocal of that base.

For Example:

 8^-1 = 1 / 8  

15^-1 = 1 / 15 

27^-1 = 1 / 27

Law 4

If  we have to multiply two numbers with same base and different exponents then 

x^ax^b = x^{a + b}

x³x⁴  = (x * x * x) * (x * x * x * x)

        = (x * x * x * x * x * x * x)

        = x^{(3 + 4)}

        = x⁷

For Example:

2⁴2³  = 2^{(4 + 3)}

= 2⁷

7⁵7⁴ = 7^{(5 + 4)}

= 7⁹

(12)⁶(12)² = 12^{(6 + 2)}

                = 12⁸

 Law 5

If we have to divide two numbers with same base and different exponents then

x^a / x^b  = x^{a – b}

x⁵ / x³  = (x * x * x * x * x) / (x * x * x)

           = (x * x)= x^{(5 – 3)}

= x²

For Example:

  3⁴ / 3²  =  3^{(4 – 2)}

              = 3²

5⁸ / 5³   =  5^{(8 – 3)}

             =  5⁵

(13)⁷ / (13)⁵ = 13^{(7 – 5)}

                    = 13²

 Law 6

(x^a)^b  =  x^ab

 (x²) ³ =  (x * x)³

          = (x * x) (x * x) (x * x) 

          = (x * x * x * x * x * x) =  x^{(2 * 3)}

          = (x)⁶

For Example:

(2³)⁴ = 2^{(3 * 4)}

= 2¹²

(5²)³ = 5^{(2 * 3)}

        = 5⁶

(13⁴)⁵ = (13)^{(4 * 5)}

= 13²⁰

Law 7

If we have two numbers to multiply with different base but same exponent then

(x * y)^a = x^ay^a

(x * y)⁴ = (xy) (xy) (xy) (xy)

           = xyxyxyxy

           = xxxxyyyy = x⁴y⁴

For Example:

(5 * 4)² = 5² * 4²

(7 * 3)⁴ = 7⁴ * 3⁴

(12 * 32)⁹ = 12⁹  * 32⁹

Law 8

If we have to divide two numbers with different base but same exponent then

(x / y)^a = x^a / y ^a

(x / y)³  = (x/y)(x/y)(x/y)

            = (x * x * x) / (y * y * y)

            = x³  / y³

For Example:

(2 / 3)⁴ = 2⁴ / 3⁴

(6 / 8)² = 6²  / 8²

(15 / 27)⁸ = 15⁸  / 27⁸

Law 9

x^-a = 1 / x^a

For Example:

8 ^-2 = 1 / 8^-2 

7^-3 = 1 / 7³

15^-6 = 1 / 15⁶

Use of Exponents to Express Small Number in Standard Form

What is a Standard Form of Number ??

Many times it Happens that we encounter a number that is very small to read and write properly, so for that purpose, there’s a better way of Describing those small numbers in Standard form.

Examples:

1. Diameter of a computer chip is 0.000003m = 3 * 10^-6m
2. Mass of dust particle is 0.000000000753 kg = 7.53 * 10^-10 kg
3. The length of the shortest visible wavelength of visible light (violet) is 0.0000004 m. = 4.0 * 10^-7 m

These numbers are very small so we will convert them to standard form lets see the steps:

• Step I: Move the Decimal to the right until there’s only 1(non-zero) Digit to the left of the decimal.
• Step II: Suppose we have shifted the decimal by n place to right then multiply the remaining number by 10^-n.

Examples:

• 0.000000000753 =  7.53 * 10^-10
• 0.0000004           =  4 * 10^-7
• 0.0000000894     =  8.94 * 10^-8
• 0.00000000052   =  5.2 * 10^-10

Applications and Uses of Exponents

1. Scientific Notation: Exponents are essential in expressing very large or very small numbers compactly in scientific notation, which is critical in fields like physics, chemistry, and engineering. For example, the speed of light in a vacuum is about 3×1083×108 meters per second, conveniently expressed using exponents.
2. Population Growth and Decay: Exponential models, which rely heavily on exponents, are used to predict population growth in biology, radioactive decay in physics, and the spread of diseases in epidemiology. For instance, the number of bacteria in a culture might grow exponentially according to the model 𝑃(𝑡)=𝑃0×𝑒𝑘𝑡P(t)=P0​×ekt, where 𝑃0P0​ is the initial population and 𝑘k is a growth constant.
3. Compound Interest: In finance, exponents are used to calculate compound interest, where the amount of money grows at a rate proportional to the current amount. The formula 𝐴=𝑃(1+𝑟/𝑛)𝑛𝑡A=P(1+r/n)nt shows how an initial investment 𝑃P grows at an interest rate 𝑟r compounded 𝑛n times per year over 𝑡t years.
4. Physics and Engineering: Exponents are used in formulas that describe physical laws, such as the inverse-square law, which states that the intensity of certain forces (like gravity or light) decreases as the square of the distance from the source increases. This is expressed as 𝐼∝1𝑟2I∝r21​.
5. Computer Science: In algorithms and complexity, exponential functions describe the complexity of certain algorithms. For example, an algorithm with exponential time complexity, denoted as 𝑂(2𝑛)O(2n), means the time to complete the task doubles with each additional element in the input data set.
6. Music Theory: Exponents are used in the mathematics of sound to describe frequencies. The frequency of notes in a musical scale, particularly in tuning and temperament systems, often involves geometric progressions that use exponents.
7. Earth Science and Meteorology: Exponential equations model many natural phenomena, such as cooling rates of geological materials and pressure changes with altitude in the atmosphere.