Properties of Powers: Definition, Rules and Examples

Properties of Powers with Natural Exponents

In this section we consider powers with positive natural exponent. Let \(a,b\in\mathbb{R}\) and let \(m,n\in\mathbb{N}^*\). The properties of powers allow us to turn products, quotients, and compound powers into simpler forms.

Each property must be applied while respecting the conditions of existence of the expressions involved. In particular, whenever quotients appear, the denominators must be different from zero.

Product of powers with the same base

The product of two powers with the same base is a power having the same base and, as its exponent, the sum of the exponents:

\[ a^m\cdot a^n=a^{m+n}. \]

Indeed, by the definition of a power,

\[ a^m=\underbrace{a\cdot a\cdot \ldots \cdot a}_{m \text{ times}}, \qquad a^n=\underbrace{a\cdot a\cdot \ldots \cdot a}_{n \text{ times}}. \]

Multiplying the two powers together yields a product consisting of \(m+n\) factors, each equal to \(a\):

\[ a^m\cdot a^n = \underbrace{a\cdot a\cdot \ldots \cdot a}_{m+n \text{ times}} = a^{m+n}. \]

Quotient of powers with the same base

If \(a\neq 0\) and \(m\geq n\), the quotient of two powers with the same base is a power having the same base and, as its exponent, the difference of the exponents:

\[ \frac{a^m}{a^n}=a^{m-n}. \]

The condition \(a\neq 0\) is required because \(a^n\) appears in the denominator.

To justify the formula, let us write the two powers as repeated products:

\[ \frac{a^m}{a^n} = \frac{\underbrace{a\cdot a\cdot \ldots \cdot a}_{m \text{ times}}} {\underbrace{a\cdot a\cdot \ldots \cdot a}_{n \text{ times}}}. \]

Since \(a\neq 0\), we may cancel \(n\) equal factors from the numerator and the denominator. There remain \(m-n\) factors equal to \(a\), so

\[ \frac{a^m}{a^n} = \underbrace{a\cdot a\cdot \ldots \cdot a}_{m-n \text{ times}} = a^{m-n}. \]

The case \(m<n\) requires the introduction of negative exponents and will be correctly interpreted in the dedicated section below.

Power of a power

The power of a power is a power having the same base and, as its exponent, the product of the exponents:

\[ (a^m)^n=a^{mn}. \]

Indeed, raising \(a^m\) to the power \(n\) means multiplying \(a^m\) by itself \(n\) times:

\[ (a^m)^n = \underbrace{a^m\cdot a^m\cdot \ldots \cdot a^m}_{n \text{ times}}. \]

Each factor \(a^m\) contains \(m\) factors equal to \(a\). Repeating this block \(n\) times, we obtain in all \(mn\) factors equal to \(a\):

\[ (a^m)^n = \underbrace{a\cdot a\cdot \ldots \cdot a}_{mn \text{ times}} = a^{mn}. \]

Power of a product

The power of a product is the product of the powers of the individual factors:

\[ (ab)^n=a^n b^n. \]

Indeed,

\[ (ab)^n = \underbrace{(ab)\cdot(ab)\cdot \ldots \cdot(ab)}_{n \text{ times}}. \]

Using the commutative and associative properties of multiplication of real numbers, we may group together all the factors equal to \(a\) and all the factors equal to \(b\):

\[ (ab)^n = \underbrace{a\cdot a\cdot \ldots \cdot a}_{n \text{ times}} \cdot \underbrace{b\cdot b\cdot \ldots \cdot b}_{n \text{ times}} = a^n b^n. \]

Power of a quotient

If \(b\neq 0\), the power of a quotient is the quotient of the powers of the numerator and the denominator:

\[ \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}. \]

Indeed,

\[ \left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b}\cdot\frac{a}{b}\cdot \ldots \cdot\frac{a}{b}}_{n \text{ times}}. \]

Multiplying together the numerators, and separately the denominators, we obtain

\[ \left(\frac{a}{b}\right)^n = \frac{\underbrace{a\cdot a\cdot \ldots \cdot a}_{n \text{ times}}} {\underbrace{b\cdot b\cdot \ldots \cdot b}_{n \text{ times}}} = \frac{a^n}{b^n}. \]

Here too the condition \(b\neq 0\) is essential, since the quotient \(\displaystyle \frac{a}{b}\) must itself be defined.

Powers with Zero Exponent

Having defined powers with positive natural exponent, it is natural to ask whether a meaning can also be assigned to a power with exponent zero.

The definition of \(a^0\) is not chosen arbitrarily: it must be compatible with the properties of powers already established for positive natural exponents.

Let \(a\neq 0\). For every \(n\in\mathbb{N}^*\), the quotient

\[ \frac{a^n}{a^n} \]

equals \(1\), because the numerator and the denominator are equal and different from zero:

\[ \frac{a^n}{a^n}=1. \]

On the other hand, if we wish to preserve the property of the quotient of powers with the same base, we must have

\[ \frac{a^n}{a^n}=a^{n-n}=a^0. \]

Comparing the two equalities, we see that, for consistency, we must set

\[ a^0=1 \qquad \text{for every } a\neq 0. \]

The condition \(a\neq 0\) is essential. Indeed, if \(a=0\), the quotient \(\displaystyle \frac{a^n}{a^n}\) becomes \(\displaystyle \frac{0}{0}\), which is undefined.

For this reason, within the context of the algebraic properties of powers, the expression \(0^0\) is left undefined.

The definition \(a^0=1\) allows the properties of powers to continue to hold even when the zero exponent appears. For example, if \(a\neq 0\) and \(m\in\mathbb{N}^*\), then

\[ a^m\cdot a^0=a^m\cdot 1=a^m=a^{m+0}. \]

Powers with Negative Integer Exponents

Having introduced the zero exponent, we may further extend the definition of a power to negative integer exponents.

Once again, the definition is not arbitrary: it is chosen so that the properties of powers continue to hold even when the exponents are no longer restricted to the natural numbers.

Let \(a\neq 0\) and let \(n\in\mathbb{N}^*\). The power with base \(a\) and exponent \(-n\) is defined by setting

\[ a^{-n}=\frac{1}{a^n}. \]

In other words, raising a nonzero number to a negative exponent means taking the reciprocal of the corresponding power with positive exponent.

The condition \(a\neq 0\) is indispensable, because the reciprocal of \(a^n\) is defined only when \(a^n\neq 0\).

The reason for this definition is the following. If we want the property of the product of powers with the same base to continue to hold, we must have

\[ a^n\cdot a^{-n}=a^{n+(-n)}=a^0. \]

Since \(a^0=1\), it must therefore follow that

\[ a^n\cdot a^{-n}=1. \]

This means precisely that \(a^{-n}\) must be the reciprocal of \(a^n\), that is,

\[ a^{-n}=\frac{1}{a^n}. \]

This definition also allows us to correctly interpret the quotient of powers with the same base in the case where the exponent of the numerator is smaller than that of the denominator.

Indeed, if \(a\neq 0\) and \(m,n\) are nonnegative integers with \(m<n\), then \(n-m>0\) and

\[ \frac{a^m}{a^n} = \frac{1}{a^{n-m}}. \]

By the definition of negative exponent,

\[ \frac{1}{a^{n-m}}=a^{-(n-m)}. \]

Since

\[ -(n-m)=m-n, \]

we obtain

\[ \frac{a^m}{a^n}=a^{m-n}. \]

In this way the property

\[ \frac{a^m}{a^n}=a^{m-n} \]

remains valid even when \(m<n\), provided that \(a\neq 0\).

More generally, if \(a\neq 0\), the properties of powers extend to integer exponents. For instance, for \(h,k\in\mathbb{Z}\) we have

\[ a^h\cdot a^k=a^{h+k}. \]

Powers with Rational Exponents

Having defined powers with integer exponent, we may extend the notion of power further to rational exponents.

In this section we consider primarily the case \(a>0\), which is the natural setting in which powers with Rational Exponents behave regularly and retain all the fundamental properties of powers.

Let \(a>0\) and let \(q\in\mathbb{N}^*\). The power with exponent \(\displaystyle \frac{1}{q}\) is defined by setting

\[ a^{\frac{1}{q}}=\sqrt[q]{a}. \]

This definition is consistent with the property of the power of a power. Indeed, if we want the identity

\[ \left(a^{\frac{1}{q}}\right)^q=a^{\frac{1}{q}\cdot q}=a \]

to continue to hold, then \(a^{\frac{1}{q}}\) must be the positive number that, when raised to the \(q\)-th power, gives \(a\). By definition, this number is the principal \(q\)-th root of \(a\).

More generally, if \(p\in\mathbb{Z}\) and \(q\in\mathbb{N}^*\), we define

\[ a^{\frac{p}{q}}=\left(\sqrt[q]{a}\right)^p. \]

Since \(a>0\), we also have \(\sqrt[q]{a}>0\), so the expression is defined even when \(p\) is negative.

When \(a>0\), the same quantity can equivalently be written in the form

\[ a^{\frac{p}{q}}=\sqrt[q]{a^p}. \]

Indeed, for positive bases, the integer powers and the arithmetic roots under consideration are always defined, and the two expressions

\[ \left(\sqrt[q]{a}\right)^p \qquad \text{and} \qquad \sqrt[q]{a^p} \]

represent the same number.

For example,

\[ 16^{\frac{3}{4}}=\left(\sqrt[4]{16}\right)^3=2^3=8. \]

If the rational exponent is negative, we again use the definition of a power with negative integer exponent:

\[ a^{-\frac{p}{q}}=\frac{1}{a^{\frac{p}{q}}}, \qquad a>0. \]

For example,

\[ 8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{\left(\sqrt[3]{8}\right)^2} = \frac{1}{2^2} = \frac{1}{4}. \]

With this definition, the properties of powers extend to rational exponents. The verification is obtained by writing the rational exponents as fractions with a common denominator and applying the properties already established for powers and roots.

In particular, for \(a>0\) and for \(r,s\in\mathbb{Q}\), the following formulas hold:

\[ a^r\cdot a^s=a^{r+s}, \qquad \frac{a^r}{a^s}=a^{r-s}, \qquad \left(a^r\right)^s=a^{rs}. \]

The restriction \(a>0\) avoids ambiguities and special cases arising from zero or negative bases. For instance, if \(a=0\), powers with positive rational exponent can be defined in many cases, whereas those with negative exponent are undefined. If instead \(a<0\), the situation requires further distinctions, and not all the properties remain valid without additional conditions.

The extension of powers to real exponents requires more advanced tools related to the concept of a limit and is treated elsewhere. On this page we restrict ourselves to natural, integer, and rational exponents.

Examples Using the Properties of Powers

Let us look at some examples of the application of the properties of powers. These examples show how to apply the rules in an orderly fashion, distinguishing between powers with the same base, powers of products, powers of quotients, and powers with negative or rational exponent.

Example 1. Let us simplify the expression

\[ a^5\cdot a^3\cdot b^2\cdot b^4. \]

We group the powers with the same base and add the exponents:

\[ a^5\cdot a^3\cdot b^2\cdot b^4 = a^{5+3}\cdot b^{2+4} = a^8b^6. \]

Hence

\[ a^5\cdot a^3\cdot b^2\cdot b^4=a^8b^6. \]

Example 2. Let us simplify the expression

\[ (a^3b^2)^4. \]

We first apply the property of the power of a product, and then the property of the power of a power:

\[ (a^3b^2)^4 = (a^3)^4(b^2)^4 = a^{3\cdot 4}b^{2\cdot 4} = a^{12}b^8. \]

Therefore

\[ (a^3b^2)^4=a^{12}b^8. \]

Example 3. Let us simplify the expression

\[ a^5\cdot a^0, \]

assuming \(a\neq 0\). Since \(a^0=1\), we obtain

\[ a^5\cdot a^0=a^5\cdot 1=a^5. \]

Example 4. Let us simplify the expression

\[ \frac{a^6b^8}{a^2b^3}, \]

assuming \(a\neq 0\) and \(b\neq 0\). We separate the powers with the same base:

\[ \frac{a^6b^8}{a^2b^3} = \frac{a^6}{a^2}\cdot\frac{b^8}{b^3}. \]

We now subtract the exponents:

\[ \frac{a^6}{a^2}\cdot\frac{b^8}{b^3} = a^{6-2}b^{8-3} = a^4b^5. \]

Hence

\[ \frac{a^6b^8}{a^2b^3}=a^4b^5. \]

Example 5. Let us simplify the expression

\[ \left(\frac{a^3b^5}{ab^2}\right)^2, \]

with \(a\neq 0\) and \(b\neq 0\). We first simplify the quotient inside the parentheses:

\[ \frac{a^3b^5}{ab^2} = a^{3-1}b^{5-2} = a^2b^3. \]

At this point we square the result:

\[ \left(a^2b^3\right)^2 = (a^2)^2(b^3)^2 = a^4b^6. \]

Therefore

\[ \left(\frac{a^3b^5}{ab^2}\right)^2=a^4b^6. \]

Example 6. Let us simplify the expression

\[ 8^{-\frac{2}{3}}. \]

The exponent is a negative rational number. We first rewrite the power as the reciprocal of the power with positive exponent:

\[ 8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}}. \]

We now use the definition of a power with rational exponent:

\[ 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4. \]

Hence

\[ 8^{-\frac{2}{3}} = \frac{1}{4}. \]