Roots and fractional powers: reversing multiplication

A square panel has area 16 m². Its side is 4 m because $4\cdot4=16$ . A cube of volume 27 m³ has edge 3 m because $3\cdot3\cdot3=27$ . Taking a root recovers a number from its power, but an even power requires a choice of sign.

We use number sets, powers and domain restrictions. The formulas $S=s^2$ for a square’s area and $V=s^3$ for a cube’s volume multiply two and three equal lengths respectively.

Area = side × side; volume = edge × edge × edge. Lengths are positive. These are separate diagrams; the cube uses schematic perspective, and sizes are not compared between drawings.

One radical symbol, two possible numbers

The principal square root $\sqrt a$ is the nonnegative number whose square is $a$ . A real result requires $a\ge0$ . Thus $\sqrt{16}=4$ , not $\pm4$ , and $\sqrt0=0$ .

Asking which numbers satisfy $x^2=16$ is different. There are two: 4 and −4, since both squares equal 16. A negative panel length is invalid, but the negative number does solve the numerical equation. Principal roots, OpenStax §1.3.

Consequently $\sqrt{x^2}=|x|$ . The absolute value $|x|$ is the distance from $x$ to zero: $x$ when $x\ge0$ , and $-x$ when $x<0$ . For example, $\sqrt{(-5)^2}=5$ , not −5. Squaring erases the original sign; the radical does not automatically recover it.

Even and odd indices

In $\sqrt[n]a$ , the integer $n\ge2$ is the root’s index. The result raised to the $n$ th power gives $a$ ; for even $n$ we choose the nonnegative result.

An even index requires a nonnegative radicand. An odd index permits any real number: $\sqrt[3]{-27}=-3$ because $(-3)^3=-27$ . Odd powers preserve the sign, so there is no ambiguity here. A negative box volume has no physical meaning; a negative cube root is an algebraic operation.

Simplifying radicals

For $a,b\ge0$ , $\sqrt{ab}=\sqrt a\sqrt b$ : the product on the right is nonnegative and its square is $ab$ . For a quotient, require $a\ge0,b>0$ : $\sqrt{a/b}=\sqrt a/\sqrt b$ . Hence $\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2$ .

Simplify first, then collect matching radicals: $\sqrt{18}+\sqrt8=3\sqrt2+2\sqrt2=5\sqrt2$ . No corresponding sum rule exists: $\sqrt{9+16}=5$ , whereas $\sqrt9+\sqrt{16}=7$ . Signs matter too: $\sqrt{(-1)(-1)}=1$ , but the separate $\sqrt{-1}$ terms are undefined over the reals.

A fractional exponent records the same operation

For $a>0$ , integer $m$ and integer $n\ge2$ , define:

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

The denominator specifies the root and the numerator the power. For example, $16^{3/4}=(\sqrt[4]{16})^3=2^3=8$ , and $16^{-1/2}=1/\sqrt{16}=1/4$ .

Exponent laws remain valid for positive bases: $a^pa^q=a^{p+q}$ , $(a^p)^q=a^{pq}$ and $(ab)^p=a^pb^p$ for $a,b>0$ and rational $p,q$ . For base zero we permit positive exponents, but neither zero nor negative ones. Negative bases need separate analysis: an odd root’s existence does not give unrestricted permission to use every law.

Branch · A transformationAn additional use of the difference of squares.

Removing a radical from a denominator

Multiply numerator and denominator by the conjugate $\sqrt3-1$ :

\frac1{\sqrt3+1}=\frac{\sqrt3-1}{(\sqrt3+1)(\sqrt3-1)}=\frac{\sqrt3-1}{2}.

Conjugates differ in the sign between terms. The multiplier is nonzero, preserving the value. This changes form, not accuracy: both expressions are exact.

A root recovers a quantity from its power under a specified sign convention. Check both radicands and denominators before transformations; a shorter formula does not cancel those conditions.