Powers and Roots

Exponents are repeated multiplication, but the real magic is what happens when the exponent stops being a whole number. Once you accept $a^{1/2}$ as a meaningful symbol, you have a tool that compresses the entire universe of polynomial growth, exponential decay, scientific notation, and the geometry of similar shapes. This section unifies them: powers and roots are the same operation, just with the exponent above or below $1$.

Whole-number powers and the five rules

For a real $a$ and positive integer $n$, define

Three rules drop out of this definition immediately:

To extend exponents to zero and negatives, we INSIST that these rules keep holding. From $a^m \cdot a^0 = a^{m+0} = a^m$ we are forced to set $a^0 = 1$ (for $a \neq 0$). From $a^n \cdot a^{-n} = a^{n + (-n)} = a^0 = 1$ we get $a^{-n} = \dfrac{1}{a^n}$.

From powers to roots

The $n$th root of $a \geq 0$ is the unique non-negative real $b$ with $b^n = a$. Notation: $b = \sqrt[n]{a}$. Existence follows from the completeness property of §3.1; uniqueness follows from the positivity of $\mathbb{R}$. For $n = 2$ we drop the index: $\sqrt{a} = \sqrt[2]{a}$.

Rational exponents unify the two ideas

If we want the rule $(a^{1/n})^n = a^{(1/n) \cdot n} = a^1 = a$ to hold, then $a^{1/n}$ MUST mean "the $n$th root of $a$." This forces the definition:

With this definition, the five rules of exponents extend automatically to all rational exponents — addition becomes addition of exponents whether they are integers or fractions.

The grapher makes the inverse relationship visible

(Slide $n$ in the widget. The curves $y = x^n$ (steep) and $y = x^{1/n}$ (shallow) are reflections of each other across the line $y = x$. That reflection IS the inverse-function relationship between a power and its root.)

Square-root algebra

For $a, b \geq 0$: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ and $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$ (with $b > 0$). These distribute over multiplication and division only — never addition.

Try it

• Simplify $\dfrac{x^5 \cdot x^{-2}}{x^3}$ using the addition rule for exponents.
• Evaluate $8^{2/3}$ by writing it as $\left(8^{1/3}\right)^2$.
• In the widget, fix $n = 3$. Where does the curve $y = x^{1/3}$ live for negative $x$? Compare with $n = 4$.
• Simplify $\sqrt{72}$ by extracting the largest perfect-square factor.

Try it in code

A trap to watch for

For even $n$, the equation $x^n = a$ with $a > 0$ has two real solutions: $\sqrt[n]{a}$ and $-\sqrt[n]{a}$. The symbol $\sqrt[n]{a}$ refers to the positive root only — this is a convention, but it is universal. So $\sqrt{9} = 3$, NOT $\pm 3$. The equation $x^2 = 9$ has solutions $x = \pm 3$, but the function $\sqrt{\cdot}$ returns only the positive value. The fix: keep these straight by always asking "am I solving an equation or applying a function?"

What you now know

You can extend exponents from positive integers all the way to negative integers and rationals, and you understand why these extensions are forced by the exponent rules rather than chosen. You can compute, simplify, and graph $a^x$ for any rational $x$. The next section uses these new powers to write and solve inequalities.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 3, §3 — powers and roots, defined cleanly and rigorously.
• Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage. Chapter 1 — exponent and root functions as the prelude to logarithms.
• Niven, I. (1961). Numbers: Rational and Irrational. Random House (New Mathematical Library, MAA). Chapter 4 — rigorous treatment of nth roots of irrationals.