Laws of Exponents for Real Numbers

The laws of exponents, also known as the rules of exponents, describe how to simplify and manipulate expressions involving exponents. These laws provide a foundation for understanding and working with exponential functions, which are crucial in various areas of mathematics, science, and engineering. This section focuses on exponents with real numbers, including integral and rational exponents, and positive real bases.

A base raised to an exponent represents repeated multiplication. For instance, $2^3$ (2 to the power of 3) means 2 multiplied by itself three times: $2 \times 2 \times 2 = 8$. Understanding these rules makes simplifying and solving complex exponential expressions much easier.

Formulae

Let $a$ and $b$ be positive real numbers, and let $m$ and $n$ be real numbers. The following laws of exponents apply:

Product of Powers: $a^m \cdot a^n = a^{m+n}$
Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
Power of a Power: $(a^m)^n = a^{m \cdot n}$
Power of a Product: $(ab)^m = a^m \cdot b^m$
Power of a Quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$
Zero Exponent: $a^0 = 1$ (where $a \neq 0$)
Negative Exponent: $a^{-n} = \frac{1}{a^n}$
Rational Exponent: $a^{\frac{1}{n}} = \sqrt[n]{a}$ (where $n$ is a positive integer)
General Rational Exponent: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ (where $n$ is a positive integer)

Examples

Here are a few examples illustrating the use of the laws of exponents:

Example-1: Simplify $3^2 \cdot 3^4$.

Using the product of powers rule: $3^2 \cdot 3^4 = 3^{2+4} = 3^6 = 729$

Example-2: Simplify $\frac{5^7}{5^3}$.

Using the quotient of powers rule: $\frac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$

Common mistakes by students

Adding Bases instead of multiplying: Forgetting the product of powers rule and incorrectly thinking $a^m \cdot a^n = (a+a)^{m+n}$.
Incorrectly applying the Power of a Product rule: Forgetting to apply the exponent to each factor inside the parentheses, such as $(2x)^2 = 2x^2$ instead of $4x^2$.
Misunderstanding negative exponents: Thinking $a^{-n}$ is always negative, instead of remembering it represents $\frac{1}{a^n}$.
Difficulty with rational exponents: Struggling to understand the relationship between rational exponents and radicals. Confusing $a^{\frac{1}{2}}$ with $\frac{1}{a^2}$ or similarly with other rational exponents.

Real Life Application

The laws of exponents are fundamental in several real-world applications:

Compound Interest: Calculating the growth of investments over time. The formula $A = P(1 + r)^t$ uses exponents.
Population Growth: Modeling population increases, often described by exponential functions.
Radioactive Decay: Describing the decay of radioactive substances, which follows an exponential decay pattern.
Computer Science: Used extensively in computer memory and storage calculations, as well as algorithm analysis.
Finance: Valuation of options, calculating the present value of future cash flows and calculating depreciation.

Fun Fact

The term “exponent” comes from the Latin word “exponere,” which means “to put forth” or “to display.” The concept of exponents has been used for millennia; it was especially important in early astronomy.