Let \( a \neq 0 \) and let \( n \in \mathbb{N} \). The \( n \)-th power of \( a \), denoted by \( a^n \), is defined as the product of \( a \) multiplied by itself \( n \) times. In formulas, this product is expressed as:
\[ a^n := \underbrace{a \cdot \ldots \cdot a}_{n \text{ times}} \]
The number \( a \) is called the base of the power, and \( n \) is the exponent of the power.
Table of Contents
Properties of Exponents
Let \( a \) and \( b \) be real numbers different from zero, and let \( m \) and \( n \) be integers. Exponents have the following fundamental properties:
Product of powers with the same base:
The product of two powers with the same base is a power with the same base and an exponent equal to the sum of the exponents:
\[ a^m \cdot a^n = a^{m+n} \]
By definition:
\[ a^m = \underbrace{a \cdot a \cdot \dots \cdot a}_{m \text{ times}} \quad , \quad a^n = \underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}} \]
Therefore, multiplying the two powers:
\[ a^m \cdot a^n = \underbrace{a \cdot a \cdot \dots \cdot a}_{m \text{ times}} \cdot \underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}} = \underbrace{a \cdot a \cdot \dots \cdot a}_{m+n \text{ times}} = a^{m+n} \]
Division of powers with the same base:
The result of the division of two powers with the same base is a power with the same base and an exponent equal to the difference of the exponents.
\[ \frac{a^m}{a^n} = a^{m-n} \quad \text{with } a \neq 0 \]
By definition:
\[ \frac{a^m}{a^n} = \frac{\underbrace{a \cdot a \cdot \dots \cdot a}_{m \text{ times}}}{\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}}} = \underbrace{a \cdot a \cdot \dots \cdot a}_{(m-n) \text{ times}} = a^{m-n}. \]
Power of a power:
The power of a power is a power with the same base and an exponent equal to the product of the exponents:
\[ (a^m)^n = a^{m \cdot n} \]
By definition:
\[ (a^m)^n = \underbrace{a^m \cdot a^m \cdot \dots \cdot a^m}_{n \text{ times}} = \underbrace{(a \cdot a \cdot \dots \cdot a)}_{m \cdot n \text{ times}} = a^{m \cdot n}. \]
Product of powers with different bases but the same exponent:
The power of a product is the product of the powers of the individual factors:
\[ (a \cdot b)^n = a^n \cdot b^n \]
By definition:
\[ (a \cdot b)^n = \underbrace{(a \cdot b) \cdot (a \cdot b) \cdot \dots \cdot (a \cdot b)}_{n \text{ times}} = (\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}}) \cdot (\underbrace{b \cdot b \cdots b}_{n \text{ times}}) = a^n \cdot b^n. \]
Quotient of powers with different bases but the same exponent:
The power of a quotient is the quotient of the powers of the numerator and the denominator:
\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad \text{with } b \neq 0 \]
By definition:
\[ \left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b} \cdot \frac{a}{b} \cdot \dots \cdot \frac{a}{b}}_{n \text{ times}} = \frac{\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}}}{\underbrace{b \cdot b \cdot \dots \cdot b}_{n \text{ times}}} = \frac{a^n}{b^n}. \]
Power with a fractional exponent:
By definition, the expression \( a^{\frac{n}{m}} \) represents the \( m \)-th root of \( a^n \), i.e.:
\[ a^{\frac{n}{m}} = \sqrt[m]{a^n} \quad \text{with } a \geq 0, \, m > 0 \]
This definition ensures that the general property of exponents is respected. For example:
\[ a^{\frac{n}{m}} \cdot a^{\frac{p}{q}} =: a^{\frac{n}{m} + \frac{p}{q}} \]
Powers with negative exponents:
A number raised to a negative exponent is equal to the reciprocal of the power with a positive exponent:
\[ a^{-n} = \frac{1}{a^n} \quad \text{with } a \neq 0 \]
Power with Exponent Zero
When we extend a definition (in this case, powers) to new cases (such as the zero exponent), we want the properties that hold in the known cases to continue to hold in the new cases as well.
For \( a \neq 0 \) and positive exponents, we know the fundamental property holds:
\[ a^m \cdot a^n = a^{m+n} \]
To maintain consistency with the division of powers, we define for \( n > 0 \):
\[ a^{-n} = \frac{1}{a^n} \]
Consider any number \( n \). By the properties of exponents, it must hold:
\[ a^n \cdot a^{-n} = a^{n+(-n)} = a^0 \]
But we also know that:
\[ a^n \cdot a^{-n} = 1 \]
Therefore, we must conclude that:
\[ a^0 = 1 \]
Hence, for any \( a \neq 0 \), it holds:
\[ a^0 = 1 \]
Exercises on Properties of Exponents
Exercise 1. Simplify: \( a^5 \cdot a^3 \cdot b^2 \cdot b^4 \)
Solution. We apply the product of powers property for the same base by adding the exponents:
\begin{align} a^5 \cdot a^3 \cdot b^2 \cdot b^4 &= a^{5+3} \cdot b^{2+4} \\ &= a^8 \cdot b^6 \end{align}
Result: \( a^8 \cdot b^6 \).
Exercise 2. Simplify \( (a^3 \cdot b^2)^4 \).
Solution. We apply the power of a product property, raising each factor to the new exponent:
\[ \begin{align*} (a^3 \cdot b^2)^4 &= (a^3)^4 \cdot (b^2)^4 \\ &= a^{3 \cdot 4} \cdot b^{2 \cdot 4} \\ &= a^{12} \cdot b^8 \end{align*} \]
Result: \( a^{12} \cdot b^8 \).
Exercise 3. Simplify:
\[ \frac{a^6 \cdot b^8}{a^2 \cdot b^3} \]
Solution. We use the quotient of powers property, subtracting the exponents:
\begin{align} \frac{a^6 \cdot b^8}{a^2 \cdot b^3} &= \frac{a^6}{a^2} \cdot \frac{b^8}{b^3} \\ &= a^{6-2} \cdot b^{8-3} \\ &= a^4 \cdot b^5 \end{align}
Result: \( a^4 \cdot b^5 \).
Exercise 4. Simplify:
\[ \left(\frac{a^3 \cdot b^5}{a \cdot b^2}\right)^2 \]
Solution. First, we simplify the terms inside the parentheses, then apply the exponent:
\[ \begin{align} \frac{a^3 \cdot b^5}{a \cdot b^2} &= \frac{a^3}{a} \cdot \frac{b^5}{b^2} \\ &= a^{3-1} \cdot b^{5-2} \\ &= a^2 \cdot b^3 \end{align} \]
Now, we apply the exponent to the simplified result:
\[ \begin{align} \left(a^2 \cdot b^3\right)^2 &= (a^2)^2 \cdot (b^3)^2 \\ &= a^{2 \cdot 2} \cdot b^{3 \cdot 2} \\ &= a^4 \cdot b^6 \end{align} \]
Result: \( a^4 \cdot b^6 \).
Exercise 5. Simplify:
\[ \frac{(a^3 \cdot b^2)^2 \cdot b^4}{a^4 \cdot b^5} \]
Solution. We start by computing the power of the numerator:
\[ \begin{align} (a^3 \cdot b^2)^2 &= (a^3)^2 \cdot (b^2)^2 \\ &= a^{3 \cdot 2} \cdot b^{2 \cdot 2} \\ &= a^6 \cdot b^4 \end{align} \]
Adding the \( b^4 \) term to the numerator:
\begin{align} a^6 \cdot b^4 \cdot b^4 &= a^6 \cdot b^{4+4} \\ &= a^6 \cdot b^8 \end{align}
Now, we simplify the fraction:
\begin{align} \frac{a^6 \cdot b^8}{a^4 \cdot b^5} &= \frac{a^6}{a^4} \cdot \frac{b^8}{b^5}, \\ &= a^{6-4} \cdot b^{8-5}, \\ &= a^2 \cdot b^3. \end{align}
Result: \( a^2 \cdot b^3 \).