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Properties of Roots



Key Terms
Even Index Radicals
Radical Notation
Odd Index Radicals
Principal Root Convention
The Identity for Even Roots
The Identity for Odd Roots
Domain and Range by Index
Sign Behavior Summary
Connection to Rational Exponents
Even vs Odd at a Glance
Properties of Roots FAQ



Even and Odd

The index of a radical determines its behavior. Even-index radicals accept only non-negative radicands and produce only non-negative outputs. Odd-index radicals accept any real number and preserve sign.

This distinction shapes everything — which inputs are valid, which outputs are possible, whether absolute values appear when simplifying, and how radical functions behave. Every property on this page traces back to whether the index is even or odd.

Key Terms

Radical Components

Index— determines whether the radical is even or odd, controlling domain and sign behavior
Radicand— the expression under the radical; must be non-negative for even indices
Principal Root— the non-negative value returned by even-index radicals, ensuring the radical behaves as a function

Related Concepts

Radical Function— domain and range depend directly on index parity
Rational Exponent— equivalent notation where domain restrictions carry over

Formulas Used on This Page

Even Root Identity— ann=∣a∣\sqrt[n]{a^n} = |a| (nn even)
Odd Root Identity— ann=a\sqrt[n]{a^n} = a (nn odd)

See All Algebra Definitions →

See All Algebra Formulas →


Even Index Radicals

Square roots, fourth roots, sixth roots — any radical with an even index — require non-negative radicands in the real number system.

16=4,814=3,646=2\sqrt{16} = 4, \quad \sqrt[4]{81} = 3, \quad \sqrt[6]{64} = 2


No real number, when raised to an even power, produces a negative result. Positive numbers raised to even powers give positives. Negative numbers raised to even powers also give positives. Zero raised to any positive power gives zero.

Therefore −9\sqrt{-9} has no real value. No real number squares to −9-9.

−9is not a real number\sqrt{-9} \quad \text{is not a real number}


Even-index radicals also produce only non-negative outputs. The principal root convention guarantees this: 25=5\sqrt{25} = 5, not −5-5. Both 5 and −5-5 square to 25, but the radical symbol returns only the non-negative root.

This restriction affects the domain of radical functions with even index: only non-negative inputs are allowed. It also determines when complex numbers become necessary.
x^2 and x^4 are never negative, so no real x gives -9xy−4−3−2−11234−12−9−6−3369121518y = −9: never reachedy = x²y = x⁴even powers are never below 0No real number squares to −9, so √(−9) is not a real number
Why even roots refuse negatives

The graphs of x² and x⁴ touch 0 at the origin and stay at or above it everywhere else. A square root of −9 would be a number whose square is −9, a point where one of these curves meets the dashed level −9, and there is none. The same holds for every even power, so every even-index root of a negative number fails to be real.

x⁴ hugs the axis more tightly near 0 but has the same floor: nothing below 0.

Radical Notation

Notation

Radical Notation

The three parts of the radical, the bar that is secretly a parenthesis, and the sign that says two things at once.
The exponent spelling a1/na^{1/n} and the bridge between the notations — rational exponents.
an\sqrt[n]{a}
The n-th root of a; root a, for the square root
Three parts: the index nn in the crook, the radicand aa under the bar, and the hook joining them. The index 22 is never written — a bare a\sqrt{a} is the square root. The LaTeX reference puts the index in brackets: \sqrt[n]{a}, and the mathematical keyboard inserts the sign with the radicand slot already waiting — square, cube and fourth roots each have their own key.
CasesIndex parity is the whole story on this page — even in Even Index Radicals above, odd in Odd Index Radicals below; domain, sign, and convention all key off it.
Also writtena1/na^{1/n} — the exponent spelling, bridged at rational exponents. Older British texts call an irrational root a “surd”.
Do not confuseA both-roots symbol. The glyph returns one number only — the non-negative one for even indices, per Principal Root Convention below; the other root needs the sign written explicitly.
9+16\sqrt{9 + 16}
The root of the whole quantity, nine plus sixteen
The overbar — the vinculum — is a grouping symbol: an invisible pair of parentheses. Everything beneath it is rooted as one block, so 9+16=25=5\sqrt{9+16} = \sqrt{25} = 5.
CasesIn print the bar's reach is exact; in handwriting an under-extended bar silently shrinks the radicand — the sloppiest common source of radical errors.
Also writtenDescartes fused the hook with the vinculum in La Géométrie (1637); before that, radicals grouped with dots or not at all.
Do not confuseA distributable operation. 9+16=5\sqrt{9+16} = 5 while 9+16=7\sqrt{9} + \sqrt{16} = 7 — the root does not distribute over sums, and the bar is precisely what marks the difference.
±\pm · ∓\mp
Plus or minus; minus or plus
Two statements on one line: x=±4x = \pm 4 unpacks to x=4x = 4 or x=−4x = -4 — unpack before computing. It appears the moment an even root meets an equation, as in Principal Root Convention below. The mathematical keyboard carries both marks.
Cases∓\mp tracks the opposite choice inside the same formula — cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B: top sign goes with top sign. The mark's most famous job is the quadratic formula on quadratic equations.
Also writtenThe solution-set form x∈{−4,4}x \in \{-4, 4\}, when the compression is unwanted.
Do not confuseMeasurement uncertainty. In the sciences 5.0±0.15.0 \pm 0.1 names an interval, not two values — same glyph, unrelated job.

Odd Index Radicals

Cube roots, fifth roots, seventh roots — any radical with an odd index — accept all real numbers as radicands.

273=3,−273=−3,−325=−2\sqrt[3]{27} = 3, \quad \sqrt[3]{-27} = -3, \quad \sqrt[5]{-32} = -2


A negative number raised to an odd power remains negative. Therefore every negative number has a real odd root.

(−3)3=−27⟹−273=−3(-3)^3 = -27 \quad \Longrightarrow \quad \sqrt[3]{-27} = -3


Odd-index radicals preserve sign. Positive radicands yield positive roots. Negative radicands yield negative roots. Zero yields zero.

There is no ambiguity requiring a principal root convention for odd indices. Every real number has exactly one real cube root, one real fifth root, and so on. The radical simply returns that unique value.

This means radical functions with odd index have domain all real numbers. No restrictions, no need for complex numbers to handle negative inputs.
x^3 takes every real value exactly once: -27 only at x = -3xy−4−3−2−11234−36−27−18−99182736y = −27y = 27y = x³∛(−27) = −3∛27 = 3one crossing per level: one real cube rootEach level meets y = x³ once, so every real number has one cube root
Every level, one cube root

The curve y = x³ rises through every height, from far below the axis to far above it. The dashed level −27 meets it exactly once, at x = −3, and the level 27 exactly once, at x = 3. One crossing for every level is why every real number, negative ones included, has exactly one real cube root.

Compare section 1: a parabola never meets a level below 0, a cubic meets every level once.

Principal Root Convention

For even-index radicals, multiple real numbers satisfy the defining equation. Both 4 and −4-4 are square roots of 16, since both square to 16.

The principal root convention resolves this ambiguity: the radical symbol denotes the non-negative root.

16=4\sqrt{16} = 4


This is not a choice between equally valid answers. It is a definition that makes the radical symbol unambiguous. Without it, 16\sqrt{16} would have two values, and expressions like 16+1\sqrt{16} + 1 would be meaningless — which value is intended?

When both roots are needed, explicit notation indicates this:

x2=16⟹x=±16=±4x^2 = 16 \quad \Longrightarrow \quad x = \pm\sqrt{16} = \pm 4


The ±\pm symbol signals that both positive and negative roots apply. This notation appears frequently when solving radical equations.

Odd-index radicals need no such convention. Each real number has exactly one real odd root, so no ambiguity exists.

The root calculator returns the principal root, following this convention.

The Identity for Even Roots

A critical identity governs even-index radicals of powers:

x2=∣x∣\sqrt{x^2} = |x|


This is not xx. It is the absolute value of xx.

When x=5x = 5: 52=25=5=∣5∣\sqrt{5^2} = \sqrt{25} = 5 = |5|. Correct.

When x=−5x = -5: (−5)2=25=5=∣−5∣\sqrt{(-5)^2} = \sqrt{25} = 5 = |-5|. Also correct.

If the identity were x2=x\sqrt{x^2} = x, then (−5)2\sqrt{(-5)^2} would equal −5-5. But square roots are non-negative by the principal root convention. The result must be 5, not −5-5.

The general form:

Even Root Identity
ann=∣a∣(n even)\sqrt[n]{a^n} = |a| \quad (n \text{ even})
Learn more about this formula: Even Root Identity →


This absolute value appears when simplifying radicals with variables. Whenever an even root extracts a variable raised to an odd power, absolute value may be required.

x6=∣x3∣=∣x∣3\sqrt{x^6} = |x^3| = |x|^3


When the resulting exponent is even, the absolute value is unnecessary since even powers are already non-negative:

x4=x2(no absolute value needed)\sqrt{x^4} = x^2 \quad \text{(no absolute value needed)}
The graph of sqrt(x^2) is the V of |x|, not the line y = xxy−5−4−3−2−112345−5−4−3−2−1123456y = x would give −5 herey = √(x²) = |x|√((−5)²) = 5√(5²) = 5for x < 0, √(x²) = −x√(x²) = |x|: the root of a square is never negative
The graph of √(x²)

Plotting √(x²) gives a V, not a line. On the right it agrees with y = x. On the left, squaring removes the sign and the root returns the positive value: x = −5 gives √25 = 5, not −5. The dashed line shows where y = x would have gone. The V is the graph of |x|, which is the identity √(x²) = |x|.

For ∛(x³) there is no fold: the graph is the whole line y = x.

The Identity for Odd Roots

Odd-index radicals behave more simply:

Odd Root Identity
ann=a(n odd)\sqrt[n]{a^n} = a \quad (n \text{ odd})
Learn more about this formula: Odd Root Identity →


No absolute value appears. The odd root returns the original value, preserving sign.

83=2,−83=−2\sqrt[3]{8} = 2, \quad \sqrt[3]{-8} = -2


x33=xfor all real x\sqrt[3]{x^3} = x \quad \text{for all real } x


This makes simplifying radicals with odd index more straightforward. Variables come out without absolute value concerns.

x93=x3\sqrt[3]{x^9} = x^3


y155=y3\sqrt[5]{y^{15}} = y^3


The distinction between even and odd indices is essential when variables are involved. Assuming all variables are positive removes the absolute value issue for even indices — a common simplifying assumption stated explicitly in problems. Without that assumption, absolute values must be tracked.

Domain and Range by Index

The index determines the domain and range of radical functions.

For even index nn:

f(x)=xnf(x) = \sqrt[n]{x}


Domain: [0,∞)[0, \infty) — only non-negative inputs allowed.

Range: [0,∞)[0, \infty) — only non-negative outputs produced.

For odd index nn:

f(x)=xnf(x) = \sqrt[n]{x}


Domain: (−∞,∞)(-\infty, \infty) — all real numbers allowed.

Range: (−∞,∞)(-\infty, \infty) — all real numbers produced.

When the radicand is an expression, solve for valid inputs:

f(x)=x−3f(x) = \sqrt{x - 3}


The radicand x−3x - 3 must be non-negative: x−3≥0x - 3 \geq 0, so x≥3x \geq 3.

Domain: [3,∞)[3, \infty).

g(x)=x−33g(x) = \sqrt[3]{x - 3}


No restriction on the radicand for odd index.

Domain: (−∞,∞)(-\infty, \infty).

These domain considerations carry into operations and equations involving radicals.
Function form Domain Range
n√x   (n even) [0, ∞) [0, ∞)
n√x   (n odd) (−∞, ∞) (−∞, ∞)
√(x − 3) [3, ∞) [0, ∞)
∛(x − 3) (−∞, ∞) (−∞, ∞)

Sign Behavior Summary

The sign of a radical depends on the radicand and the index.

When a>0a > 0:

an>0for all n\sqrt[n]{a} > 0 \quad \text{for all } n


Positive radicands always yield positive roots, regardless of index.

When a=0a = 0:

0n=0for all n\sqrt[n]{0} = 0 \quad \text{for all } n


Zero yields zero.

When a<0a < 0 and nn is odd:

an<0\sqrt[n]{a} < 0


Negative radicands yield negative roots for odd indices.

When a<0a < 0 and nn is even:

anis not a real number\sqrt[n]{a} \quad \text{is not a real number}


No real even root of a negative number exists. This is where complex numbers enter.

Understanding sign behavior prevents errors in simplifying and operations. It also explains why extraneous solutions arise when solving radical equations — squaring both sides can introduce negative values where only non-negative values are valid.
Case Result of n√a Example
a > 0, any n positive √16 = 4,   ∛27 = 3
a = 0, any n zero √0 = 0,   ∛0 = 0
a < 0, n odd negative (preserves sign) ∛(−27) = −3
a < 0, n even not a real number √(−9) requires complex numbers

Connection to Rational Exponents

Every property of radicals translates to rational exponents.

an=a1/n\sqrt[n]{a} = a^{1/n}


The domain restrictions are identical. For even nn, the expression a1/na^{1/n} requires a≥0a \geq 0. For odd nn, any real aa is permitted.

The absolute value identity becomes:

(xn)1/n=∣x∣when n is even(x^n)^{1/n} = |x| \quad \text{when } n \text{ is even}


(xn)1/n=xwhen n is odd(x^n)^{1/n} = x \quad \text{when } n \text{ is odd}


The radical rules correspond directly to exponent laws:

abn=(ab)1/n=a1/n⋅b1/n=an⋅bn\sqrt[n]{ab} = (ab)^{1/n} = a^{1/n} \cdot b^{1/n} = \sqrt[n]{a} \cdot \sqrt[n]{b}


Working in exponent notation often simplifies algebraic manipulation, especially when combining expressions with different indices. Converting to a common denominator in the exponents enables operations that would be awkward in radical form.

The properties established here — domain restrictions, sign behavior, absolute value requirements — apply equally whether the expression is written with a radical or a fractional exponent.

Even vs Odd at a Glance

The properties covered above all reduce to a single distinction: whether the index is even or odd. The table below collects every parity-driven property in one place — radicand restrictions, sign behavior, domain, range, the radical-of-a-power identity, and what happens when the radicand goes negative. It works as a study card and as a quick check when working through any expression involving radicals.
Property Even index (n = 2, 4, 6, …) Odd index (n = 3, 5, 7, …)
Allowed radicand (real values) a ≥ 0 only any real a
Sign of result always ≥ 0 (principal root) matches sign of a
Domain of f(x) = n√x [0, ∞) (−∞, ∞)
Range of f(x) = n√x [0, ∞) (−∞, ∞)
Identity n√(xn) |x| x
Negative radicand result not real — use complex numbers real, negative

Properties of Roots FAQ

Why does x2=∣x∣\sqrt{x^2} = |x| while x33=x\sqrt[3]{x^3} = x?

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Index parity decides it. An even root must return a non-negative value under the principal root convention, so (−5)2=5\sqrt{(-5)^2} = 5 rather than −5-5, and the bars are what force that. An odd root carries no such restriction and keeps the sign, so (−2)33=−2\sqrt[3]{(-2)^3} = -2. In general ann=∣a∣\sqrt[n]{a^n} = |a| for even nn and aa for odd nn.Read more →

When does an even root need absolute value bars?

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Only when the exponent you extract comes out odd. x6=∣x3∣=∣x∣3\sqrt{x^6} = |x^3| = |x|^3 needs them, because x3x^3 is negative whenever xx is. But x4=x2\sqrt{x^4} = x^2 needs none, since an even power is already non-negative and the bars would change nothing. Check the parity of the exponent you are left with, not the one you started with.Read more →

Why doesn't a square root show its index?

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Because 22 is the default and is left off by convention, so a bare a\sqrt{a} means the square root. Every other index is written in the crook: a3\sqrt[3]{a}, a4\sqrt[4]{a}. In LaTeX the index goes in square brackets before the radicand, and the mathematical keyboard gives square, cube and fourth roots their own keys.Read more →

Does ±\pm mean the same thing in science as in algebra?

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No. In algebra x=±4x = \pm 4 is two statements compressed onto one line, unpacking to x=4x = 4 or x=−4x = -4, and you unpack before computing. In the sciences 5.0±0.15.0 \pm 0.1 names an interval of uncertainty, a range rather than two values. Same glyph, unrelated job. Its partner ∓\mp tracks the opposite choice inside one formula.Read more →