Laws of Exponents: 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.    

What are Exponents

An exponent is an expression in mathematics that defines the number of times a number needs to be multiplied. For example, in the expression xⁿ, x is referred to as the base, while n is referred to as the exponent. This merely means that you will multiply x by itself n times. Exponents are such a help in working with very large or very small numbers because they are simplified. They are widely applicable in many fields, from science and engineering, for the expression of values in a compact form and calculation purposes.

Let's Discuss each of the laws in more detail

Product of Powers

Law: x^a × x^b = x^a+b

Example: 5² × 5³ = 5²⁺³ = 5⁵

Explanation: When multiplying two powers with the same base, add their exponents.

Quotient of Powers

Law: x^a / x^b = x^a-b

Example: 5⁵ / 5² = 5^5-2 = 5³

Explanation: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Power of a Power

Law: (x^a)^b = x^axb

Example: (5²)³ = 5^2x3 = 5⁶

Explanation: When raising a power to another power, multiply the exponents.

Power of a Product

Law: (xy)^a = x^ay^a

Example: (5X3)³ = 5³ x 3³ = 25 X 9

Explanation: When raising a product to a power, raise each factor to the power.

Power of a Quotient

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

Example: (5/2)³ = 5³ / 2³ = 125 / 8

Explanation: When raising a quotient to a power, raise both the numerator and the denominator to the power.

Zero Exponent

 Law: x⁰ = 1

Example: 5^{0 = 1}

^{Explanation: Any non-zero number raised to the power of zero is 1.}

Negative Exponent

Law: x^-a = 1 / x^a

Example: 5^-2 = 1 / 5² = 1 / 25

Explanation: A negative exponent represents the reciprocal of the base raised to the absolute value of the exponent.

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 ?(?)=?0×???P(t)=P0​×ekt, where ?0P0​ is the initial population and ?k 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 ?=?(1+?/?)??A=P(1+r/n)nt shows how an initial investment ?P grows at an interest rate ?r compounded ?n times per year over ?t 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 ?∝1?2I∝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 ?(2?)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.

Laws of Exponents

1. Product of Powers:

When you multiply two powers that have the same base, you add their exponents.

Example: 𝑥^𝑎 × 𝑥^𝑏 becomes 𝑥^𝑎+𝑏

2. Quotient of Powers:

When you divide two powers that have the same base, you subtract the exponent of the denominator from the exponent of the numerator.

Example: x^a / x^b becomes x^a-b

3. Power of a Power:

When raising a power to another power, you multiply the exponents.

Example: (x^a)^b becomes x ^a*b

4. Power of a Product:

When raising a product to a power, each factor is raised to that power.

Example: (xy)^a becomes x^a * y^a

5. Power of a Quotient:

When raising a quotient to a power, both the numerator and the denominator are raised to that power.

Example: (x/y)^a becomes x^a / y^a

6. Zero Exponent:

Any non-zero number raised to the power of zero equals 1.

Example: x⁰ =1

Applications of Exponents in Real Life

1. Scientific Notation:

The speed of light is about 3 × 10^8 meters per second, which is very large can be expressed in compact form. For instance, such enormous numbers or very small numbers can easily be expressed in this form.

2. Population Growth and Radioactive Decay:

In biology, bacterial population growth, grows exponentially. Similarly in physics, radioactive decay of carbon and many more elements are expressed in exponents

3. Compound Interest:

Essentially, exponents are used to compute compound interest in finance. The formula says, an investment will grow as interest is added to the original principal. Hence, it grows in value exponentially.

4. Physics and Engineering:

Many physical laws involve exponents, and one of the classic examples is the inverse-square law, which states how the intensity of forces like gravity drops as the distance from the source increases.

5. Computer Science:

Exponents describe the complexity of some tasks, such as algorithms. For example, one with exponential time complexity would double the amount of time needed to complete with every additional element in the input data set.

6. Music Theory

The exponents are useful in understanding sound frequencies for music. The frequency of the notes in a musical scale more often than not forms patterns described by exponents.

7. Earth Science and Meteorology:

Some exponential equations model things such as the cooling of geological materials or the change in pressure with altitude in the atmosphere.

Practice Problems on Laws of Exponents

1. Simplify: 2³×2⁴

2. Evaluate: 7⁰

3. Simplify: (3²)³

4. Express 0.00004 in scientific notation.

5. Calculate the compound interest for a principal of $1000 at an interest rate of 5% compounded annually for 3 years.

Summary

Exponents are powerful mathematical tools used to simplify expressions, solve equations, and describe various real-life phenomena. Understanding the laws of exponents and their applications can significantly enhance problem-solving skills in mathematics, science, and engineering.

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Article Tags :

• Mathematics
• School Learning
• Class 8
• Maths-Class-8