Exponents and Powers

Exponents and Powers are unleashing the magic of Mathematics. The most fascinating dialect is the realm of exponents and powers. This world of mathematical wizardry involves tiny superscripts that have the power to transform numbers, equations, and even our understanding of the universe itself. Let's embark on a journey to explore the enchanting world of exponents and powers.

Exponents

Exponential notation is a way to represent the repeated multiplication of a number by itself.
Here's how they work:

→ Base: The base is the number you want to multiply.

→ Exponent: The exponent (or power) is a small number written above and to the right of the base. It tells you how many times to multiply the base by itself.

For example, in the expression 53, the base is 5 and the exponent is 3. This means you multiply 5 by itself three times: (5 × 5 × 5) which equals 125.

Laws of Exponents

The laws of exponents are as follows:

1. Multiplying Numbers with Exponents:

If two non-zero rational numbers x and y and raise them to the same integer power (m), then

2. Combining Powers with the Same Base:

If a non-zero rational number x is raised to the power of m and raised to the power of n, then

3. Reciprocals and Negative Exponents:

→ When you have a number with a negative exponent, it is the same as taking its reciprocal (1 divided by that number) with a positive exponent:

→ If you have a fraction with a negative exponent, it is like flipping the fraction and making the exponent positive:

4. Any Number Raised to the Power of 0:

→ Any number (except zero) raised to the Power of 0 is always equal to 1.

x⁰ = 1

5. Writing Small Numbers in Standard Form:

→ When you have very small numbers, you can express them in standard form using negative exponents.

→ The standard form of a number is a representation where the number is expressed as the product of a decimal number between 1.0 and 10.0 (including 1.0 and excluding 10.0) multiplied by a power of 10.

Example: The mass of an electron is 0.00000000000000000000000000000091 kg.

The mass of an electron expressed in standard form is shown as

Mass of an electron  = 0.00000000000000000000000000000091 kg
= 91 × 10⁻³² kg
= 9.1 × 10⁻³¹ kg    

This makes it easier to work with really small or large numbers.