Let \( a \neq 0 \) and let \( n \in \mathbb{N} \). The \( n \)-th power of \( a \), denoted by the symbol \( a^n \), is defined as the product of \( a \) by itself \( n \) times. Mathematically, 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, \( n \) is the exponent of the power.
Contents
- Properties of Powers
- Power with Zero Exponent
- Powers with Negative Exponent
- Powers with Fractional Exponent
- Exercises on the Properties of Powers
Properties of Powers
Let \( a \) and \( b \) be non-zero real numbers, and let \( m \) and \( n \) be integers. Powers have the following fundamental properties:
Product of powers with the same base:
The product of two powers with the same base is a power that has the same base and the sum of the exponents as the exponent:
\[ a^m \cdot a^n = a^{m+n} \]
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 that has the same base and the difference of the exponents as the exponent.
\[ \frac{a^m}{a^n} = a^{m-n} \quad \text{with } a \neq 0 \]
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 that has the same base and the product of the exponents as the exponent:
\[ (a^m)^n = a^{m \cdot n} \]
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 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:
\[ (a \cdot b)^n = \underbrace{(a \cdot b) \cdot (a \cdot b) \cdot \dots \cdot (a \cdot b)}_{n \text{ times}} = (\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ times}}) \cdot (\underbrace{b \cdot b \cdot \dots \cdot b}_{n \text{ times}}) = a^n \cdot b^n. \]
Quotient of powers with different bases but same exponent:
The power of a quotient is the quotient of the powers of the numerator and 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 Zero Exponent
When we extend a definition (in this case powers) to new cases (such as the zero exponent), we want the properties already valid in the known cases to continue to hold in the new cases as well.
For \(a \neq 0\) and for positive exponents, we know that the fundamental property holds:
\[ a^m \cdot a^n = a^{m+n} \]
Consider any natural number \(n\). By the property of powers, it must hold that:
\[ a^n \cdot a^{-n} = a^{n+(-n)} = a^0 \]
But we also know that:
\[ a^n \cdot a^{-n} = a^n \cdot \frac{1}{a^n} = 1 \]
Therefore, by the transitive property \( a^0 = 1 \).
This definition keeps all the properties of powers consistent. For example:
\[ a^m \cdot a^0 = a^m \cdot 1 = a^m = a^{m+0} \]
\[ \frac{a^m}{a^m} = a^{m-m} = a^0 = 1 \]
The definition \(a^0 = 1\) is not arbitrary, but is the only one that guarantees the consistency of the algebraic rules of powers.
Powers with Negative Exponent
A number raised to a negative exponent equals the reciprocal of the power with positive exponent:
\[ a^{-n} = \frac{1}{a^n} \quad \text{with } a \neq 0 \]
This definition derives from the need to maintain consistency with the property of division of powers. If we want \(\frac{a^m}{a^n} = a^{m-n}\) to always hold, then for \(m < n\) we obtain a negative exponent in the result.
By definition of division:
\[ \frac{a^m}{a^n} = \frac{1}{a^{n-m}} = \frac{1}{a^{-(m-n)}} = a^{-(n-m)} = a^{m-n} \]
This definition ensures that all properties of powers extend consistently to negative exponents. For example:
\[ a^m \cdot a^{-n} = a^m \cdot \frac{1}{a^n} = \frac{a^m}{a^n} = a^{m-n} = a^{m+(-n)} \]
Powers with Fractional Exponent
To extend the definition of power to fractional exponents, we must maintain consistency with the properties already established for integer exponents.
By definition, the expression \(a^{\frac{n}{m}}\) indicates the \(m\)-th root of \(a^n\), that is:
\[ a^{\frac{n}{m}} = \sqrt[m]{a^n} \quad \text{with } a \geq 0, \, m > 0 \]
This definition can be equivalently written as:
\[ a^{\frac{n}{m}} = (\sqrt[m]{a})^n \]
The definition is not arbitrary but derives from the need to preserve the fundamental property of powers. If we want \(a^x \cdot a^y = a^{x+y}\) to continue to hold, then for the exponent \(\frac{1}{m}\) it must necessarily hold that:
\[ (a^{\frac{1}{m}})^m = a^{\frac{1}{m} \cdot m} = a^1 = a \]
This means that \(a^{\frac{1}{m}}\) is that number which, when raised to the power \(m\), returns \(a\). By definition of root, this is exactly \(\sqrt[m]{a}\).
All properties of powers extend naturally to fractional exponents:
\[ a^{\frac{p}{q}} \cdot a^{\frac{r}{s}} = a^{\frac{p}{q} + \frac{r}{s}} = a^{\frac{ps + qr}{qs}} \]
The definition ensures that the general property of powers is respected and maintains the consistency of the entire algebraic structure.
Exercises on the Properties of Powers
Exercise 1. Simplify: \( a^5 \cdot a^3 \cdot b^2 \cdot b^4 \)
Solution. We apply the property of the product of powers with the same base, 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 property of powers to the product, 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 property of the division of powers with the same base, 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. We start by simplifying the terms inside the parentheses, then apply the power to the result:
\[ \begin{align} \frac{a^3 \cdot b^5}{a \cdot b^2} &= \frac{a^3}{a} \cdot \frac{b^5}{b^2} \\ &= a^{3-1} \cdot b^{5-2} \\ &= a^2 \cdot b^3 \end{align} \]
Now, we apply the power 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 calculating the power in the numerator:
\[ \begin{align} (a^3 \cdot b^2)^2 &= (a^3)^2 \cdot (b^2)^2 \\ &= a^{3 \cdot 2} \cdot b^{2 \cdot 2} \\ &= a^6 \cdot b^4 \end{align} \]
We add the term \( b^4 \) 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 quotient:
\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: \( a^2 \cdot b^3 \).