Radicand — Definition, Examples & Common Mistakes

Q: What is the difference between the radicand and the index?

In $\sqrt[n]{x}$, the radicand $x$ is the value under the radical whose root you are finding, while the index $n$ is the small number written above the radical that tells you which root to take. If no index is written, it is understood to be 2 (a square root).

Radicand

The number under the Square root symbol (√) with 3 as the radicand: √3 (radical) symbol. That is, a number which is having its square root taken (or cube root, 4th root, 5th root, nth root, etc.). For example, 3 is the radicand in Square root of 5, where 5 is the radicand under the radical symbol .

Key Formula

\sqrt[n]{x}

Where:

$x$ = The radicand — the number or expression under the radical sign
$n$ = The index — indicates which root is being taken (2 for square root, 3 for cube root, etc.)
$\sqrt{\phantom{x}}$ = The radical symbol itself

Example

Problem: Identify the radicand in each expression: (a) √49, (b) ∛(8x²), (c) ⁴√(2a + 5).

Step 1: In

\sqrt{49}

, the number sitting under the radical symbol is 49. The radicand is 49.

\sqrt{\mathbf{49}} \quad \Rightarrow \quad \text{radicand} = 49

Step 2: In

\sqrt[3]{8x^2}

, the entire expression under the radical is

8x^2

. Even though it contains a variable, the whole thing is the radicand.

\sqrt[3]{\mathbf{8x^2}} \quad \Rightarrow \quad \text{radicand} = 8x^2

Step 3: In

\sqrt[4]{2a+5}

, the expression

2a + 5

is under the radical. The radicand is

2a + 5

\sqrt[4]{\mathbf{2a+5}} \quad \Rightarrow \quad \text{radicand} = 2a + 5

Answer: The radicands are (a) 49, (b) 8x², and (c) 2a + 5. The radicand is always everything underneath the radical symbol, whether it is a single number, a variable expression, or a combination of both.

Another Example

Problem: Simplify √72 by rewriting the radicand as a product of a perfect square and another factor.

Step 1: The radicand is 72. Find the largest perfect square that divides 72. Since

72 = 36 \times 2

, and 36 is a perfect square, use this factorization.

\sqrt{72} = \sqrt{36 \times 2}

Step 2: Apply the product rule for radicals: separate the radicand into two radicals.

\sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2}

Step 3: Evaluate

\sqrt{36} = 6

. The remaining radicand, 2, has no perfect-square factors, so it stays under the radical.

\sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}

Answer:

\sqrt{72} = 6\sqrt{2}

. Simplifying a radical often involves factoring the radicand to extract perfect squares (or perfect cubes, etc.).

Frequently Asked Questions

Can the radicand be negative?

It depends on the index. For even roots (square root, 4th root, etc.), a negative radicand has no real-number result — you would need imaginary numbers. For odd roots (cube root, 5th root, etc.), the radicand can be negative because, for example,

\sqrt[3]{-8} = -2

What is the difference between the radicand and the index?

\sqrt[n]{x}

, the radicand

x

is the value under the radical whose root you are finding, while the index

n

is the small number written above the radical that tells you which root to take. If no index is written, it is understood to be 2 (a square root).

Radicand vs. Index (of a radical)

Both are parts of a radical expression

\sqrt[n]{x}

. The radicand (

x

) is the quantity under the radical sign — the input whose root you want. The index (

n

) is the small number tucked into the notch of the radical sign that specifies which root is being taken (square root, cube root, etc.). Changing the radicand changes what number you are taking the root of; changing the index changes which type of root you compute.

Why It Matters

Knowing what the radicand is helps you communicate clearly when simplifying radicals, solving radical equations, and discussing domains of functions. When you simplify

\sqrt{72}

, you factor the radicand — so recognizing it is the first step. In algebra and calculus, restrictions on a function like

f(x) = \sqrt{x - 3}

come directly from requiring the radicand

x - 3

to be non-negative.

Common Mistakes

Mistake: Confusing the radicand with the index.

Correction: In

\sqrt[3]{27}

, the index is 3 (it tells you 'cube root') and the radicand is 27 (the number whose root you find). The index sits outside the radical notch; the radicand sits under the bar.

Mistake: Thinking the radicand must be a single number.

Correction: The radicand can be any expression — a variable, a polynomial, or a product. In

\sqrt{x^2 + 9}

, the entire expression

x^2 + 9

is the radicand, not just

x^2

or 9 alone.

Related Terms

Radical — The symbol and notation containing the radicand
Square Root — Root with index 2; radicand must be ≥ 0
Cube Root — Root with index 3; radicand can be negative
nth Root — General root notation using any index
Argument of a Function — The radicand serves as the argument of a root function
Extraneous Solution — Can arise when solving equations involving radicands