Nth Root Calculator
Calculate the nth root of any number (ⁿ√x). Supports even and odd roots, real roots, radical expressions, and step-by-step solutions.
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What Is a Nth Root Calculator?
An nth root of a number x is a number r which, when raised to the power n, yields x: r^n = x. Here, n is a positive integer called the index or degree of the radical.
How to Use This Calculator
- Enter the radicand (x) and root degree (n ≥ 1).
- View the principal real root or complex root indication.
- Review the exponential relationship x^(1/n).
Nth Root & Fractional Exponent Relationship
r = \sqrt[n]{x} \iff r^n = x, \quad \sqrt[n]{x} = x^{1/n}Taking the nth root is mathematically equivalent to raising the radicand to the power of 1/n.
Worked Example
Scenario: Find the 5th root of 32
Radicand: x = 32, Degree: n = 5
Search for r such that r⁵ = 32
Notice 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
Result: ⁵√32 = 2
Tips & Key Notes
- Odd roots of negative numbers are real and negative: ∛(-8) = -2, ⁵√(-32) = -2.
- Even roots of negative numbers have no real solution; their roots are complex numbers.
- In calculus, fractional exponents x^(1/n) make power rule differentiation much simpler.
Frequently Asked Questions
Can you take an odd root of a negative number?
Yes! Because a negative number raised to an odd power remains negative, odd roots of negative numbers exist on the real number line (e.g. ∛(-27) = -3).
What is the index of a radical?
The index is the small number n sitting in the crook of the radical sign that specifies which root is being taken.
How are roots related to exponents?
The nth root of x is identical to raising x to the reciprocal power: ⁿ√x = x^(1/n).
How do you multiply radicals with different indices?
Convert radicals to fractional exponents with a common denominator, then multiply using exponent laws.
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