Table of Contents
In mathematics, an exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression 3⁴, the base is 3, and the exponent is 4, meaning that 3 is multiplied by itself four times: 3 × 3 × 3 × 3. In this case, the exponent 4 tells us the number of times to multiply 3 together. We also refer to this as “3 raised to the power of 4.”
Exponents can be applied to various types of numbers, including whole numbers, fractions, negative numbers, and decimals. This article will discuss the concept of exponents in detail and provide you with examples to help understand their use.
What are Exponents?
The exponent of a number shows how many times the base is repeatedly multiplied. For example, 2⁴ represents 2 being multiplied by itself four times or 2 × 2 × 2 × 2. Here, 2 is the base, and 4 is the exponent or power.
Meaning of Exponents
An exponent is a way to represent large numbers by using powers. For example, when you multiply 4 by itself three times, it can be written as 4 × 4 × 4 = 4³. In this case, the base is 4, and it is raised to the power of 3, indicating that 4 is multiplied by itself three times.
To further understand exponents, consider the general form xⁿ where x is the base and n is the exponent. This means that x is multiplied by itself n times:
xⁿ = x × x × x × … × x (multiplied n times)
- Within the defined term xn,
- x itself is referred to as the ‘base’
- n is known as the ‘exponent’
- The expression xn is pronounced as ‘x raised to n’ (or) ‘x to the power of n’
Examples of Exponents
- 3 × 3 × 3 × 3 × 3 × 3 = 3⁶
- (-2) × (-2) × (-2) × (-2) = (-2)⁴
- a × a × a × a × a × a × a = a⁷
The importance of exponents lies in their ability to simplify the representation of numbers that are multiplied by themselves multiple times. Instead of writing a long multiplication, such as 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5, we can conveniently express it as 58. This concise notation makes it easier to read, write, and perform calculations, especially when dealing with large numbers.
Properties of Exponents
Exponent rules are also known as the properties of exponents. These rules simplify calculations and provide a structured approach when working with powers. Here are the basic properties of exponents:
- Product Law: aᵐ × aⁿ = aᵐ⁺ⁿ
- Quotient Law: aᵐ / aⁿ = aᵐ⁻ⁿ
- Zero Exponent Law: a⁰ = 1
- Negative Exponent Law: a⁻ⁿ = 1 / aⁿ
- Power of a Power Law: (aᵐ)ⁿ = aᵐⁿ
- Power of a Product Law: (ab)ᵐ = aᵐbᵐ
- Power of a Quotient Law: (a / b)ᵐ = aᵐ / bᵐ
Also Check – Equivalent Fractions
Negative Exponents
A negative exponent indicates how many times we need to multiply the reciprocal of the base. For example, a⁻ⁿ is equivalent to 1 / aⁿ. This means that instead of multiplying the base a, we multiply its reciprocal 1 / a a total of n times.
Negative exponents are particularly useful when dealing with fractions. Below are a few examples of expressions with negative exponents and how to convert them to positive exponents:
Examples of Negative Exponents
- 2 × 3⁻⁹ = 2 × (1 / 3⁹) = 2 / 3⁹
- 9⁻³ = 1 / 9³
- 6⁻⁷ = 1 / 6⁷
Exponents with Fractions
A fractional exponent is an exponent expressed as a fraction and represents roots such as square roots, cube roots, and other nth roots.
For example:
- If the exponent is a¹/², it corresponds to the square root of a.
- If the exponent is a¹/³, it represents the cube root of a.
Decimal Exponent
When an exponent is written in decimal form, it is called a decimal exponent. Calculating exact values for decimal exponents can be difficult, so we usually look for approximate values. One effective method to simplify decimal exponents is by converting them into fractions first.
For example, 4¹.⁵ can be rewritten as 4³/², which simplifies to 8.
Scientific Notation with Exponents
Scientific notation is a standard method for expressing very large or very small numbers using decimals and powers of ten. A number is written in scientific notation when it is represented as the product of a number between 1 and 10, multiplied by a power of 10.
- If the number is greater than 1, the exponent is positive.
- If the number is less than 1, the exponent is negative.
To convert a number into scientific notation, follow the below-given steps:
Step One: Place a decimal point after the leftmost non-zero digit. If the number has only one significant digit (excluding zeros), no decimal is needed.
Step Two: Multiply this new number by a power of 10. The exponent will equal the number of decimal places you moved.
- If you move the decimal to the left, the exponent is positive.
- If you move it to the right, the exponent is negative.
By following these steps, any number can be written in scientific notation. For example:
For example:
- 560,000 becomes 5.6 × 10⁵
- 0.00736567 becomes 7.36567 × 10⁻³
Tips and Tricks for Exponents
- Whenever a fraction has a negative exponent, find its reciprocal to make the exponent positive. For example, (a / b)⁻ᵐ = (b / a)ᵐ.
- Decimal exponents can first be solved by converting the decimal number into a fraction, similar to 2⁰.⁵ = 4¹/².
Major Forms of Exponents
There are four main types of exponents:
- Rational exponent: When square roots, cube roots, or other roots are involved, the exponent is expressed as a fraction. For example, aᵐ/ⁿ = ⁿ√aᵐ.
- Positive exponent: A positive exponent means the base is multiplied by itself the number of times indicated by the exponent. For example, 2³ = 2 × 2 × 2 = 8.
- Negative exponent: A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. For example, 2⁻³ = 1 / 2³ = 1 / 8.
- Zero exponent: Any base raised to the power of zero equals 1. For example, a⁰ = 1 (as long as a ≠ 0).
Exponents FAQs
What is the exponent also known as?
An exponent is also known as an index or power.
What is 2 raised to the power of 5?
2 to the power of 5 can be expressed as 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
What is an exponent?
An exponent refers to the number of times a number is multiplied by itself.
