The index of a radical is the small number written just above and to the left of the radical symbol (√) that tells you which root to take. If you see ³√8, the index is 3, meaning you’re looking for the cube root. If no number appears there, the index is 2 by default, giving you a square root.
Parts of a Radical Expression
A radical expression has three components that work together. The radical symbol (√) indicates you’re taking a root. The radicand is whatever sits inside the symbol, the value you’re finding the root of. The index is the small number tucked into the crook of the radical symbol, specifying which root you want.
In the expression ⁴√81, the index is 4, the radicand is 81, and the answer is 3 (because 3 × 3 × 3 × 3 = 81). The index essentially asks: “What number, multiplied by itself this many times, gives the radicand?”
Why the Index Is Sometimes Invisible
Square roots are so common that mathematicians skip writing the 2. When you see √25, the index is 2 even though nothing is written there. This is purely a shorthand convention. Every other root requires the index to be displayed: ³√ for cube roots, ⁴√ for fourth roots, ⁵√ for fifth roots, and so on.
Each index has a name. An index of 2 is a square root, an index of 3 is a cube root, and beyond that, you simply say “the nth root.” So ⁵√32 is read as “the fifth root of 32.”
How Even and Odd Indices Behave Differently
The index does more than label which root you’re taking. It controls what numbers you’re allowed to put under the radical in the first place.
When the index is even (2, 4, 6, …), the radicand must be zero or positive. You can’t take the square root of a negative number and get a real result, because no real number multiplied by itself produces a negative. The same applies to fourth roots, sixth roots, and every other even-indexed radical. Plug in a negative radicand and the result leaves the real number system entirely.
When the index is odd (3, 5, 7, …), the radicand can be any real number, including negatives. The cube root of −8 is −2, because (−2) × (−2) × (−2) = −8. Odd-indexed radicals handle negative inputs without any trouble.
This distinction matters when you’re working with functions. A square root function only accepts inputs of zero or greater, so its graph starts at a point and curves to the right. A cube root function accepts all real numbers, so its graph stretches infinitely in both directions. Even-indexed radicals produce U-shaped curves, while odd-indexed radicals produce smoothly increasing or decreasing ones.
Converting Radicals to Fractional Exponents
Every radical can be rewritten as a fractional exponent, and the index becomes the denominator of that fraction. The general rule is:
ⁿ√(x) = x^(1/n)
So √x = x^(1/2), ³√x = x^(1/3), and ⁴√x = x^(1/4). If there’s already an exponent on the radicand, it becomes the numerator. For example, ³√(x²) = x^(2/3). The exponent inside goes on top, and the index goes on the bottom.
This conversion is useful because exponent rules are often easier to work with than radical rules. Multiplying, dividing, and simplifying expressions becomes more straightforward when everything is written with fractional exponents instead of radical symbols.
Index vs. Coefficient
A common point of confusion is mixing up the index with a coefficient in front of the radical. In the expression 3√5, that 3 is a coefficient, meaning “3 times the square root of 5.” The index is still 2 (the invisible default). In ³√5, the 3 is the index, meaning “the cube root of 5.” These are completely different values: 3√5 ≈ 6.71, while ³√5 ≈ 1.71.
On paper, the distinction is visual. The index sits small and snug in the “V” of the radical symbol, while a coefficient is a full-sized number standing in front. In typed math, the index is usually written as a superscript before the radical or inside brackets, like ³√ or root[3]. If you’re reading a math problem and aren’t sure which you’re looking at, check the size and position of the number relative to the radical sign.
Practical Examples by Index
- Index 2 (square root): √49 = 7, because 7 × 7 = 49
- Index 3 (cube root): ³√27 = 3, because 3 × 3 × 3 = 27
- Index 4 (fourth root): ⁴√16 = 2, because 2 × 2 × 2 × 2 = 16
- Index 5 (fifth root): ⁵√32 = 2, because 2 × 2 × 2 × 2 × 2 = 32
As the index increases, the root value gets closer to 1 for any given radicand. The square root of 1,000,000 is 1,000, but the sixth root of 1,000,000 is 10. Higher indices “pull down” the result more aggressively, because you’re looking for a number that reaches the radicand through more repeated multiplications.
The index must always be a positive integer greater than 1. An index of 1 would just return the radicand unchanged, so it’s never used. Fractional or negative indices don’t appear in standard radical notation, though the equivalent operations can be expressed using fractional exponents.