Difference Between Power and Exponent: Complete Details

The difference between Power and Exponent is that a power expression multiplies the same number over and over again, while an exponent expression raises a number to a high power. Both terms are often used interchangeably in mathematical operations. Both play a crucial role in simplifying expression, solving equations, and working with vast or tiny numbers. Understanding the difference between power and exponent will equip you to navigate these mathematical operations with confidence.

What is Power?

In mathematics, a power also called an exponent refers to a shorthand way of writing repeated multiplication of the same number, it contains two parts.

• Base: This is the number you’re multiplying by itself.
• Exponent: This is usually a smaller number written above and to the right of the base that tells how many times to multiply the base by itself.

The entire expression, based on the exponent, is called a power. For example, 5³ (read as “five cubed”) is a power where 5 is the base and 3 is the exponent. It signifies multiplying 5 by itself three times: 5³ = 5 x 5 x 5.

What is Exponent?

Exponent, also sometimes called power, to the right of a number in math expressions. It tells you exactly how many times to multiply the number (the base) by itself. 

• Exponent: The small number written above and to the right of the base e.g., the 3 in 7³.
• Base: The number being multiplied by itself e.g., the 7 in 7³.

So, the exponent essentially acts like a shortcut, saving you from writing out the same number multiplied repeatedly. For instance, 4² read as “ 4 to the power of 2” is the same as 4 x 4.

What is the Difference Between Power and Exponent?

Here is a table summarizing the difference between Power and Exponent.

Power	Exponent
The entire expression represents repeated multiplication.	The small number is written above and to the right of the base in a power expression.
The result of multiplying the base by itself a certain number of times.	The number of times the base is multiplied by itself.
2³ (2 multiplied by itself 3 times like 2 x 2 x 2)	3 (the number of times 2 is multiplied)
Represents the complete concept of repeated multiplication.	Indicates the instruction for how many times to multiply the base.

Also Read: Difference Between Marginal Cost and Average Cost

Difference Between Power and Exponent Formulas

Here are some important formulas related to powers and exponents.

Basic Rules:

• a¹ = a (Any number to the power of 1 equals itself)
• a⁰ = 1 (Any number except 0 to the power of 0 equals 1) (where a ≠ 0 ).

Multiplication Rule:

• a^m × aⁿ = a^m+n  (Multiplying powers with the same base, add the exponents)
• a^m × b^m = (ab)^m (Power of a product, exponent applies to each base separately)

Power of Power Rule:

• (a^m)ⁿ = a^{(m x n)} (Power of a power, multiply the exponents)

Division Rule:

• a^m ÷ aⁿ = a^m-n (Dividing powers with the same base, subtract the exponents)
• a^m ÷ b^m = (a/b)^m (Power of a quotient, exponent applies to both numerator and denominator)

Negative Exponents Rule:

• a^-n = 1/aⁿ (Negative exponent, reciprocate the base and keep the exponent positive) where a ≠ 0.

NOTE: These formulas only apply when dealing with real numbers as the base and whole numbers (positive or negative) as the exponent.

Application of Difference Between Power and Exponent in Real Life

Powers and exponents pop up in many different ways in our daily lives. Here are some real-world applications.

• We use exponents to calculate areas and volumes.
• Scientists deal with both incredibly large (distance in space) and incredibly small (atoms) numbers. For instance, the distance to the sun is about 1.5 x 10¹¹ kilometres (150 billion kilometres).
• Many things in nature and society grow exponentially. For instance, bacterial populations can double every few minutes, which can be modelled using exponential functions.
• Radioactive materials decay exponentially over time. The half-life of a radioactive element tells you how long it takes for half of the material to decay, and this is expected using an exponent.
• Computer memory is measured in units like gigabytes (GB) or terabytes (TB). These are based on powers of 2.1 GB is equal to 2³⁰ bytes.
• Digital images are made up of pixels. The resolution of an image refers to the number of pixels horizontally and vertically.
• Higher resolution involves a greater number of pixels, often expressed using exponents, e.g., 1920 x 1080.

Also Read: Difference Between Parallel and Perpendicular

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