Powers

Introduction to powers

1.Powers are mathematical operations that indicate how many times a number is to be multiplied by itself. The standard notation for a power is $a^n$ where $a$ is the base and $n \in \mathbb{Z}^+$ is the exponent:

$$a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}} \qquad a \in \mathbb{R}$$

2.

From a geometric perspective, for a positive number $a$, the expression $a^2$ is the area of a square with side length $a$, while $a^3$ is the volume of a cube with edge length $a$.

3.The exponent $n$ can be positive, negative, or zero. A negative exponent indicates a reciprocal relationship between powers, expressed as follows:

$$a^{-n} = \frac{1}{a^{n}}$$

4.Raising a number to a negative exponent is therefore equivalent to taking the reciprocal of the same number raised to the corresponding positive exponent. If the exponent is fractional, for example $n = 1/m$, the power can be rewritten in radical form as follows:

$$a^{\frac{1}{m}} = \sqrt[m]{a}$$

5.This expression is the $m$-th root of $a$. By combining both concepts, a negative fractional exponent can be interpreted as the reciprocal of a root:

$$a^{-\frac{1}{m}} = \frac{1}{\sqrt[m]{a}}$$

6.This notation expresses both powers and roots with a single exponent, and it keeps the exponent rules consistent across the different cases.

7.The table below shows selected values of $a^n$ where each row corresponds to a fixed base $a$ and each column to a fixed exponent $n$.

8.

The symbol — indicates that the expression is undefined: negative exponents of zero involve division by zero, and $0^0$ is an indeterminate form.

9.If the base $a$ is negative, the sign of $a^n$ alternates according to the parity of $n$. For instance, $(-1)^n = 1$ when $n$ is even, and $(-1)^n = -1$ when $n$ is odd. This alternating behavior is particularly important in the study of sequences and limits. In addition, negative bases with fractional exponents must be treated carefully, since expressions such as $(-1)^{1/2}$ are not defined in $\mathbb{R}$.

Powers with real exponents

10.The definition of $a^n$ applies only when the exponent $n$ is a positive integer. However, the idea of repeated multiplication does not extend directly to non-integer exponents. To define powers with real exponents we adopt a different approach based on the exponential function and the natural logarithm. For any positive base $a > 0$ and real exponent $x \in \mathbb{R}$ we define:

$$a^x = e^{x \ln a}$$

11.When $x$ is a positive integer, this coincides with the usual interpretation as repeated multiplication. For rational $x$, it agrees with the radical interpretation. To verify this, let $x = p/q$ with $p, q \in \mathbb{Z}$ and $q \neq 0$. Applying the definition yields the following:

$$e^{\frac{p}{q} \ln a} = \left(e^{\ln a}\right)^{\frac{p}{q}} = a^{\frac{p}{q}}$$

12.This confirms that the exponential definition reduces to the usual radical interpretation when the exponent is rational, and the two notations are fully consistent. The need for this extension is illustrated by the expression $2^{\sqrt{2}}$. Since $\sqrt{2}$ is irrational and cannot be written as a fraction $m/n$, the radical definition does not apply. Using the definition for the real exponent, we obtain:

$$2^{\sqrt{2}} = e^{\sqrt{2} \ln 2} \approx e^{0.9803} \approx 2.665$$

13.This value is well-defined and can be approximated to any specified degree of precision. This definition also clarifies why the base must satisfy $a > 0$. If $a \leq 0$, then $\ln a$ is undefined in $\mathbb{R}$, and the extension fails. This is consistent with the earlier observation that negative bases with fractional exponents do not yield real numbers. As a consequence, for $a > 0$ the function $f(x) = a^x$ is continuous and differentiable for all real numbers.

Fundamental rules of powers

14.The following rules describe how to work with expressions involving powers. They apply to real bases and exponents under the conditions specified in each case.

15.Raising any nonzero base to the exponent zero always gives $1$. The restriction on $a$ is necessary because the expression $0^0$ is undefined and is treated as an indeterminate form.

$$a^0 = 1 \quad \text{if} \quad a \neq 0$$

16.Any power of zero always results in zero because it corresponds to the product of $n$ zeros, where $n$ is the exponent:

$$0^n = 0 \quad \text{for } n > 0$$

17.The condition $n > 0$ is necessary because $0^0$ is an indeterminate form. Depending on the context, it can be approached as a limit in two ways:

$$ \begin{align} \lim_{x \to 0^+} 0^x &= 0 \\[6pt] \lim_{x \to 0} x^0 &= 1 \end{align} $$

18.Thus, in the setting of limits, the expression $0^0$ is typically left undefined. More generally, the following expressions are indeterminate forms, meaning their values cannot be determined without additional information about the specific limit: $0^0$, $1^{\infty}$, and $\infty^0$.

19.

In combinatorics and algebra, however, $0^0$ is often defined to be $1$ by convention, because this choice makes many formulas, such as those in the binomial theorem, work consistently.

20.The product of two or more powers with the same base $a$ is again a power with base $a$, whose exponent is the sum of the exponents:

$$a^n \cdot a^m = a^{n+m}$$

21.To see why this holds, observe that $a^n$ is the product of $n$ factors equal to $a$, and $a^m$ is the product of $m$ such factors. Concatenating these two products gives a total of $n+m$ factors, which is precisely $a^{n+m}$. In formal terms:

$$ a^n \cdot a^m = \underbrace{a \cdot a \cdots a}_{n \text{ times}} \cdot \underbrace{a \cdot a \cdots a}_{m \text{ times}} = \underbrace{a \cdot a \cdots a}_{n+m \text{ times}} = a^{n+m} $$

22.This argument directly justifies the rule for positive integer exponents, since it relies on counting factors. For integer, rational, or real exponents, the same result follows from the definition $a^x = e^{x \ln a}$. Since the exponential function satisfies $e^u \cdot e^v = e^{u+v}$, we obtain:

$$a^n \cdot a^m = e^{n \ln a} \cdot e^{m \ln a} = e^{(n+m) \ln a} = a^{n+m}$$

23.The product of powers with different bases $a$ and $b$ but the same exponent $n$ is a power whose base is the product of the original bases.

$$a^{n} \cdot b^n = (ab)^n$$

24.

Consider the expression $2^3 \cdot 5^3$. Since both factors have the same exponent, they can be combined into a single power with the product of the bases as its base: $2^3 \cdot 5^3 = (2 \cdot 5)^3 = 10^3$.

25.The quotient of two powers with the same base $a$ is a power with the same base and exponent equal to the difference between the exponents.

$$\frac{a^n}{a^m} = a^{n-m}$$

26.This follows directly from the product rule established above. Dividing by $a^m$ is equivalent to multiplying by $a^{-m}$, so the quotient can be rewritten as follows: $$\frac{a^n}{a^m} = a^n \cdot a^{-m} = a^{n+(-m)} = a^{n-m}$$ The condition $a \neq 0$ is required for $a^{-m}$ to be defined.

27.The quotient of powers with different bases $a$ and $b$ but the same exponent $n$ is a power whose base is the quotient of the original bases.

$$\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$$

28.

Consider the expression $6^3/2^3$. Since both terms have the same exponent, we can rewrite the quotient as a single power whose base is the ratio of the original bases: $6^3/2^3 = \left(6/2\right)^3 = 3^3 = 27$.

29.The power of a power is a power with the same base and exponent equal to the product of the two exponents.

$$(a^m)^n = a^{m \cdot n}$$

30.

Consider the expression $(2^3)^4$. Applying the rule for a power of a power, we multiply the exponents: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

31.When a base $a$ is raised to a negative exponent, the result is the reciprocal of the same base raised to the corresponding positive exponent. $$a^{-n} = \frac{1}{a^n} \quad \text{with} \quad a \neq 0 \quad \text{and} \quad n > 0$$

32.When the exponent is a rational number of the form $\frac{m}{n}$, the power can be expressed as a radical. If the base depends on a variable, this notation may define an irrational function.

$$a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad \text{where} \quad m, n \in \mathbb{N} \quad \text{and} \quad n \neq 0$$

33.

These properties of powers allow us to transform products and quotients of powers into simpler operations on exponents, making calculations faster and more manageable in algebraic expressions.

Why is $a^0 = 1$?

34.Among the properties of exponents, the identity $a^0 = 1$ may at first seem counterintuitive, but it follows directly from the quotient rule, which states that dividing any nonzero power by itself yields $1$.

$$\frac{a^n}{a^n} = 1$$

35.Applying the quotient rule to the left-hand side we obtain:

$$\frac{a^n}{a^n} = a^{n-n} = a^0$$

36.This reasoning shows that $a^0 = 1$. The condition $a \neq 0$ is essential, since the expression $a^n/a^n$ is defined only when $a \neq 0$.

Bernoulli's inequality

37.Among the elementary results involving integer powers, one of the most useful is the inequality attributed to Jakob Bernoulli. For every real number $x \ge -1$ and every natural number $n \in \mathbb{N}$ we have:

$$ (1+x)^n \ge 1+nx $$

38.The condition $x \ge -1$ ensures that the base $1+x$ is non-negative and that the power $(1+x)^n$ is well-defined in $\mathbb{R}$ for every $n$. Equality occurs when $n = 0$, when $n = 1$, or when $x = 0$; in every other case the inequality is strict.

39.A proof proceeds by induction on $n$. For $n = 0$ both sides equal $1$, and for $n = 1$ both sides equal $1+x$. Assume that the inequality holds for some $n \ge 1$:

$$ (1+x)^n \ge 1+nx $$

40.Multiplying both sides by the non-negative quantity $1+x$ preserves the direction of the inequality, and gives:

$$ (1+x)^{n+1} \ge (1+nx)(1+x) = 1+(n+1)x+nx^2 $$

41.The term $nx^2$ is non-negative for every real $x$, so it can be discarded without violating the inequality, yielding:

$$ (1+x)^{n+1} \ge 1+(n+1)x $$

42.This establishes the statement for $n+1$ and completes the induction.

43.

The inequality extends to real exponents $r \ge 1$ when $x \ge -1$ and to $r \le 0$ when $x > -1$; it reverses for $0 < r < 1$ when $x \ge -1$. These generalizations rely on the convexity and concavity properties of the power function $t \mapsto t^r$ and are typically established using differential calculus.

44.Bernoulli's inequality provides sharp estimates with minimal effort. A classical application is the proof that the sequence $a_n = \left(1+\frac{1}{n}\right)^n$ is monotonically increasing. The argument is based on the ratio of consecutive terms and on a single application of the inequality to control the resulting expression. Since this sequence is also bounded above, it converges, and its limit is Euler's number $e$:

$$ e = \lim_{n \to +\infty}\left(1+\frac{1}{n}\right)^n $$

45.In this sense Bernoulli's inequality is one of the elementary tools for introducing $e$ without the full machinery of analysis.

Power and exponential

46.People often confuse power with exponential functions. A power is an arithmetic operation in which a base $a$ is multiplied by itself $n$ times, where $n$ is the exponent. An exponential function, by contrast, is a function in which the variable appears in the exponent rather than in the base. It has the form:

$$f(x) = e^x \quad \text{or} \quad f(x) = a^x$$

47.where $a > 0$ and $a \neq 1$.

Hierarchy of growth rates

48.A recurring problem in mathematical analysis is comparing functions that tend to infinity as the variable becomes arbitrarily large. Among such functions, those involving logarithms, powers, and exponentials form a strict hierarchy, often called the hierarchy of growth rates or the scale of infinities. The fundamental statement is that for every $\alpha > 0$ and every $a > 1$ we have:

$$ \lim_{x \to +\infty} \frac{\log_a x}{x^{\alpha}} = 0 \qquad \text{and} \qquad \lim_{x \to +\infty} \frac{x^{\alpha}}{a^{x}} = 0 $$

49.The first limit asserts that any positive power of $x$, however small the exponent, eventually dominates the logarithm. The second states that any exponential with base greater than $1$ eventually dominates every power of $x$, however large the exponent. Combining the two statements yields a strict three-tier ordering in which logarithms grow more slowly than powers, and powers grow more slowly than exponentials.

50.

For any prescribed threshold $\varepsilon > 0$, one can find an $x_0$ such that the corresponding quotient remains below $\varepsilon$ for every $x > x_0$. The point at which dominance becomes apparent depends on the parameters $\alpha$ and $a$, but the asymptotic behavior does not.

51.Both limits can be derived from a single fundamental result by means of elementary substitutions. Setting $y = \log_a x$, so that $x = a^y$, transforms the first limit into the second up to suitable changes in the constants. The second limit then follows from the series expansion of the exponential function. For every $a > 1$ we can write $a^x = e^{x \ln a}$, and since the exponential function dominates each of its truncated Taylor polynomials, the following estimate holds for every natural number $k$:

$$ e^{u} \ge \frac{u^{k}}{k!} \qquad \forall \ u \ge 0 $$

52.Choosing $k$ larger than $\alpha$ and setting $u = x \ln a$, we obtain a lower bound on $a^x$ that grows faster than any prescribed power of $x$. Consequently the quotient $x^{\alpha}/a^{x}$ is bounded above by an expression that tends to zero, which establishes the desired limit.

53.This hierarchy simplifies asymptotic estimates. Whenever a sum or product involves terms of different growth orders, only the dominant term contributes to the leading-order behavior. For example:

$$ \lim_{x \to +\infty} \frac{2^{x}+x^{10}+\ln x}{2^{x}} = 1 $$

54.The contributions of $x^{10}$ and $\ln x$ become negligible compared with $2^{x}$ as $x$ tends to infinity, even though for moderate values of $x$ the polynomial term may still be numerically larger. The same hierarchy underlies the standard classification of algorithmic complexity, in which logarithmic, polynomial, and exponential running times are treated as distinct regimes.

Powers with complex exponents

55.The definition $a^x = e^{x \ln a}$ extends to the cases where the exponent is complex. When the exponent is purely imaginary, that is, when $x = i\theta$ with $\theta \in \mathbb{R}$, the expression $e^{i\theta}$ is well-defined provided one interprets the exponential function via its Taylor series expansion. Recall that:

$$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$$

56.and substitute $z = i\theta$. The resulting series separates into real and imaginary parts by exploiting the periodicity of the powers of $i$, yielding Euler's formula.

$$e^{i\theta} = \cos\theta + i\sin\theta$$

57.This result connects the exponential function and trigonometry, which are different expressions of the same structure over the complex numbers. Any nonzero complex number can then be written in exponential form as $z = re^{i\theta}$, where $r$ denotes the modulus and $\theta$ the argument. Setting $\theta = \pi$ gives Euler's identity:

$$e^{i\pi} + 1 = 0$$

58.This identity relates the five constants $e$, $i$, $\pi$, $1$, and $0$ in a single expression. A more detailed discussion of complex exponents, including methods for computing powers and roots of complex numbers, appears on the page complex numbers in exponential form.