What is the relationship between fractional exponents and roots?

A fractional exponent expresses the same thing as a root: a^(1/n) = ⁿ√a and a^(p/n) = ⁿ√(aᵖ) = (ⁿ√a)ᵖ. For example, 8^(2/3) = ∛(8²) = ∛64 = 4, or equivalently (∛8)² = 2² = 4. In the 'Exponent' tab, type the exponent as a fraction (e.g. 2/3) and the tool converts the result to radical form.

How can you tell if a number is a perfect square or perfect cube?

A number is a perfect square if its square root is an integer (e.g. 144 → √144 = 12), and a perfect cube if its cube root is an integer (e.g. 216 → ∛216 = 6). In practice, factor the number into primes; if every exponent is a multiple of 2 it's a perfect square, and if every exponent is a multiple of 3 it's a perfect cube. The 'Perfect square / cube checker' mini tool shows the result along with the nearest perfect powers.

Exponent & Radical Calculator | Powers, Square Roots, nth Roots & Simplify

Raise a number to a power (integer, negative, or fractional exponent), find a square root, cube root, or nth root, and simplify radical expressions (√72 = 6√2) — all in one tool, with step-by-step, exact + decimal, and scientific-notation results.

Base (a)

Can be decimal or negative (e.g. 1.5 or -3).

Exponent (n)

Enter an integer, a negative number, or a fraction like 2/3.

Root index (n)

2 = square root, 3 = cube root, 4 = 4th root…

Number under the root (a)

The radicand. Can be negative for an odd-index root.

Breaks the integer under the root into prime factors and reduces it to the simplest radical form (e.g. √72 = 6√2).

Root index (n)

Number under the root (a)

Leading coefficient (c) — optional

Simplifies c·ⁿ√a as a whole, e.g. 5√72 → 30√2. Leave at 1 to simplify √a alone.

Exponent / radical expression

Allowed: + − × ÷ ^ ( ), √, ∛, sqrt(), cbrt(), root(a,n), pi, e. Example: 2^(1/2), root(64,3). Use a period for decimals (2.5).

⚙️ Advanced — decimal places, scientific notation, step-by-step

Decimal places

Scientific notationAlso show the result as a × 10ⁿ

Step-by-step solutionShow the calculation steps

Quick answer An exponent is repeated multiplication: aⁿ = a × a × … × a. A radical is its inverse: ⁿ√a is the number whose nth power equals a. This tool calculates powers, square roots, cube roots, and nth roots, simplifies radical expressions (√72 = 6√2), and shows exact, decimal, and scientific-notation results step by step.

√2 (square root of 2)

√3 (square root of 3)

∛2 (cube root of 2)

π (pi)

⚙️ Accuracy note: Integer exponent and root calculations are performed exactly; simplification breaks the integer under the root into prime factors and pulls out full powers. Irrational results (e.g. √2) have infinite decimals and are shown rounded to your chosen number of places. Even-index roots have no real value for a negative radicand. All calculations run in your browser; no data is sent to a server.

√ Exponent & Radical Calculator — ToolPico ↗

Exponents and radicals: a short but complete guide

How exponentiation, square/cube/nth roots, negative and fractional exponents, and radical simplification work, with examples.

An exponent is a compact way of writing repeated multiplication of a base: aⁿ = a × a × … × a (n factors). A radical is the inverse operation: ⁿ√a is the number whose nth power equals a. The two are linked by the identity a^(1/n) = ⁿ√a. The calculator above does both, plus it simplifies radical expressions and evaluates mixed expressions.

How do you raise a number to a power?

Quick answerExponentiation multiplies the base by itself as many times as the exponent: aⁿ = a × a × … × a. Example: 2⁵ = 2×2×2×2×2 = 32. Special cases: a⁰ = 1 (a≠0), a¹ = a, 1ⁿ = 1. When multiplying like bases, exponents add; when dividing, they subtract: aᵐ·aⁿ = aᵐ⁺ⁿ.

What is a negative exponent?

Quick answerA negative exponent gives the multiplicative inverse of the base: a⁻ⁿ = 1 / aⁿ. Example: 2⁻³ = 1/2³ = 1/8 = 0.125. In other words, a negative exponent sends the number to the denominator and shrinks it. 0⁻ⁿ is undefined.

What is a fractional (rational) exponent?

Quick answerA fractional exponent expresses a root: a^(1/n) = ⁿ√a and a^(p/n) = ⁿ√(aᵖ). Example: 8^(2/3) = ∛(8²) = ∛64 = 4. In the "Exponent" tab, entering a fraction like 2/3 as the exponent converts the result to radical form.

How do you calculate a square root?

Quick answerA square root (√a) is the positive number whose square equals a: √a = b ⇔ b² = a. Example: √144 = 12. If the number isn't a perfect square, the result is irrational; the simplest radical form is given along with a decimal approximation (√2 ≈ 1.41421).

How do you find a cube root or an nth root?

Quick answerA cube root (∛a) is the number whose cube equals a: ∛27 = 3, ∛(−8) = −2. In general, ⁿ√a is the number whose nth power equals a, equivalent to a^(1/n). Example: ⁴√16 = 2, ⁵√32 = 2. An even-index root can't take a negative radicand (in real numbers); an odd-index root can.

How do you simplify a radical (√72 = 6√2)?

Quick answerThe number under the root is broken into prime factors, and any full powers matching the root's index are pulled outside. √72 = √(2³×3²) = √(36×2) = 6√2. For cube roots, groups of three come out: ∛54 = ∛(27×2) = 3∛2. The "Simplify" tab shows every step.

Popular exponent & radical calculations

One-click examples that jump straight to the calculator. Each link opens the matching tab and fills in the values.

Related mini calculators

Fractional exponent ↔ root conversion, perfect square/cube checking, and scientific notation — all instant, in your browser.

🔁Fractional exponent ↔ root

Converts a^(p/q) to radical form: a^(p/q) = ᵠ√(aᵖ). Simplifies the result when possible.

Base (a)

Numerator (p)

Denominator (q)

🔍Perfect square / cube checker

Checks whether a number is a perfect square, cube, or nth power, and shows the nearest perfect powers.

Number

Check index (n)

🔬Scientific notation

Shows a very large or very small number in a × 10ⁿ form alongside standard notation.

Number

Reference tables

Perfect squares, perfect cubes, powers of 2 and 10, and the core exponent & radical identities — all worth memorizing.

Perfect squares: n and n² (1–20)
n	n² (perfect square)

If a number's square root matches a value in this table, the result is an integer; otherwise it's irrational.

Perfect cubes: n and n³ (1–15)
n	n³ (perfect cube)

Cube roots of perfect cubes are integers: ∛27 = 3, ∛64 = 4. Also defined for negatives: ∛(−8) = −2.

Powers of 2 and 10
Exponent (n)	2ⁿ	10ⁿ

Powers of 2 are common in computing (bytes, KB, MB); powers of 10 are used constantly in scientific notation.

Exponent rules
Rule	Example
aᵐ · aⁿ = aᵐ⁺ⁿ	2³·2² = 2⁵ = 32
aᵐ / aⁿ = aᵐ⁻ⁿ	2⁵/2² = 2³ = 8
(aᵐ)ⁿ = aᵐˑⁿ	(2³)² = 2⁶ = 64
(a·b)ⁿ = aⁿ·bⁿ	(2·3)² = 4·9 = 36
a⁻ⁿ = 1 / aⁿ	2⁻³ = 1/8 = 0.125
a⁰ = 1 (a≠0)	7⁰ = 1
a^(1/n) = ⁿ√a	8^(1/3) = ∛8 = 2

Root (radical) rules
Rule	Example
√(a·b) = √a · √b	√72 = √36·√2 = 6√2
√(a/b) = √a / √b	√(9/4) = 3/2
ⁿ√(aᵐ) = a^(m/n)	∛(8²) = 8^(2/3) = 4
a·ⁿ√b + c·ⁿ√b = (a+c)ⁿ√b	2√3 + 5√3 = 7√3
(ⁿ√a)ⁿ = a	(√5)² = 5
ⁿ√a · ⁿ√b = ⁿ√(a·b)	√2·√8 = √16 = 4

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Exponent & radical terms glossary

Short definitions of the core terms used with exponents and radicals.

BaseThe number being multiplied by itself in an exponential expression. In aⁿ, a is the base.

Exponent (power)The number of times the base is multiplied by itself. In aⁿ, n is the exponent.

RadicandThe number under the root symbol. In ⁿ√a, a is the radicand.

Root indexThe order of the root. In ⁿ√a, n; 2 means square root, 3 means cube root.

Perfect squareA number that is the square of an integer (1, 4, 9, 16, 25…). Its square root is an integer.

Perfect cubeA number that is the cube of an integer (1, 8, 27, 64…). Its cube root is an integer.

Fractional exponentAn exponent made of a numerator and denominator. a^(p/q) converts to a root: ᵠ√(aᵖ).

Negative exponentRepresents the multiplicative inverse: a⁻ⁿ = 1/aⁿ. Sends the base to the denominator.

Irrational numberA number that can't be written as a fraction; its decimal expansion is infinite and non-repeating (e.g. √2, π).

Scientific notationWriting a number as a × 10ⁿ, where 1 ≤ |a| < 10. Example: 4200 = 4.2 × 10³.

Prime factorizationWriting a number as a product of primes (72 = 2³·3²). The basis of radical simplification.

SimplificationReducing a radical expression to its simplest form by pulling full powers out of the root (√72 = 6√2).

In-depth guides

Detailed answers to the questions people mix up most.

Why does √72 equal 6√2? Step-by-step simplification

Step 1: Break the number under the root into prime factors: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Step 2: For a square root, group the factors in pairs; every complete pair comes out of the root as a single factor. 2² is a perfect square → 2 comes out; 3² is a perfect square → 3 comes out; one leftover 2 stays inside the root.

Step 3: √72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2. Decimal value ≈ 8.485. Enter 72 in the "Simplify" tab and click Calculate to see the same steps.

How is 8^(2/3) calculated? The link between fractional exponents and roots

In a fractional exponent, the numerator (top number) is the power and the denominator (bottom number) is the root index: a^(p/q) = ᵠ√(aᵖ) = (ᵠ√a)ᵖ. For our example, 8^(2/3) can be worked two ways: power first → ∛(8²) = ∛64 = 4; or root first → (∛8)² = 2² = 4. Both give 4.

Practical tip: taking the root first usually keeps the numbers smaller and easier to work with. Try it in the "Exponent" tab by typing 2/3 as the exponent with base 8 — the tool shows both the radical form and the decimal value.

Can you take the root of a negative number? Even vs. odd root index

For an even-index root (square root, 4th root…), the radicand can't be negative, because no real number raised to an even power is negative. √(−4) is undefined in real numbers (it equals 2i in complex numbers). For an odd-index root (cube root, 5th root…), a negative radicand is fine: ∛(−8) = −2, because (−2)³ = −8.

This tool warns you if you enter a negative radicand with an even index, and correctly signs the result for an odd index.

Frequently asked questions

How do you raise a number to a power (exponentiation)?

Exponentiation means multiplying the base by itself as many times as the exponent: aⁿ = a × a × … × a (n factors). For example, 2⁵ = 2×2×2×2×2 = 32. Any nonzero number raised to the power 0 equals 1 (a⁰ = 1, a≠0), and any number raised to the power 1 equals itself. Enter the base and exponent in the "Exponent" tab for the exact integer, decimal, and scientific-notation result, step by step.

How do you calculate a square root?

The square root of a number (√a) is the positive number whose square equals it: √a = b means b² = a. For example, √144 = 12 because 12² = 144. Numbers that aren't perfect squares have irrational square roots, given as a decimal approximation (√2 ≈ 1.41421). In the "Root" tab, leave the index at 2 and enter the number — the tool returns both the simplified radical form and the decimal value.

How do you simplify a radical?

The number under the radical is broken into prime factors, and any full powers matching the root's index are pulled outside the radical. For example, √72 = √(2³×3²) = √(36×2) = 6√2. For cube roots, groups of three matching factors come out: ∛54 = ∛(27×2) = 3∛2. The "Simplify" tab shows these steps along with the prime factorization.

What is a negative exponent?

A negative exponent gives the multiplicative inverse of the base raised to the positive exponent: a⁻ⁿ = 1 / aⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. The larger the base, the smaller the result. A negative exponent of zero (0⁻ⁿ) is undefined. In the "Exponent" tab, enter a negative value (e.g. -3) in the exponent field to calculate it.

How do you find a cube root?

The cube root of a number (∛a) is the number whose cube equals it: ∛a = b means b³ = a. For example, ∛27 = 3 (3³=27), ∛64 = 4. Cube roots are also defined for negative numbers: ∛(−8) = −2. If the number isn't a perfect cube, the result is irrational. In the "Root" tab, set the index to 3.

How do you calculate an nth root?

The nth root of a number (ⁿ√a) is the number whose nth power equals it, written in exponent form as a^(1/n). For example, ⁴√16 = 2 (2⁴=16), ⁵√32 = 2. Even-index roots cannot have a negative radicand; odd-index roots can. In the "Root" tab, enter your chosen index n in the index field.

Are fractional exponents and roots the same thing?

Yes. A fractional exponent expresses a root: a^(1/n) = ⁿ√a and a^(p/n) = ⁿ√(aᵖ) = (ⁿ√a)ᵖ. For example, 8^(2/3) = ∛(8²) = 4. In the "Exponent" tab, type the exponent as a fraction (e.g. 2/3) and the tool converts the result to radical form.

How accurate are the results, and how are they computed?

Integer exponent and root calculations are done exactly; irrational results are rounded to your chosen number of decimal places (2–12). All calculation happens in your browser using standard algebra rules; no data is sent to a server. The method is evergreen and doesn't depend on data that changes year to year. Last reviewed: July 19, 2026.

How do you simplify a radical that already has a coefficient, like 5√72?

Simplify the radical part first, then multiply its pulled-out factor by the existing coefficient: √72 = 6√2, so 5√72 = 5 × 6√2 = 30√2. The "Simplify" tab has an optional "Leading coefficient" field — enter 5 there with 72 under the root to get 30√2 directly, with full step-by-step working.

How do you find a missing base or exponent when you know the result of aⁿ?

To find the base when the exponent and result are known, take the nth root of the result: a = ⁿ√result (e.g. if a³ = 27, a = ∛27 = 3). To find the exponent when the base and result are known, use logarithms: n = log(result) / log(base) (e.g. if 2ⁿ = 32, n = log(32)/log(2) = 5). In the "Exponent" tab, use the "Find base" or "Find exponent" toggle to solve either one automatically.

Methodology & sources

ToolPico's Exponent & Radical Calculator is a free, independent tool. Integer exponentiation is computed with exact integer arithmetic (BigInt); root extraction and simplification work by prime-factoring the radicand and pulling out full powers that match the root's index. Fractional exponent–root conversion is based on the identity a^(p/q) = ᵠ√(aᵖ). Irrational values are computed with IEEE-754 floating point and rounded to the selected number of places.

Basis: Standard middle/high-school algebra (exponent and radical identities). The concepts are evergreen and don't depend on rates or dates. Last updated: July 19, 2026. Results are for informational and educational purposes only.

Exponent & Radical Calculator

Exponents and radicals: a short but complete guide

How do you raise a number to a power?

What is a negative exponent?

What is a fractional (rational) exponent?

How do you calculate a square root?

How do you find a cube root or an nth root?

How do you simplify a radical (√72 = 6√2)?

Popular exponent & radical calculations

Related mini calculators

Reference tables

Add this calculator to your site (embed code)

Exponent & radical terms glossary

In-depth guides

Frequently asked questions

Methodology & sources

🔗 Add this tool to your site