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# Radicand — Definition, Symbol, and Examples

TL;DR  
The radicand is the number or expression sitting under a radical (root) sign — in 49\sqrt{49}49​, the radicand is 49. This article covers the radicand's definition, the three parts of a radical, how the radicand differs from the radical, what kinds of values a radicand can hold, and the slips students make.

## What Is a Radicand?  
The **radicand** is the number or expression written _inside_ (under) a radical sign. It is the value whose root is being taken. In 49\sqrt{49}49​, the radicand is **49**; in 273\sqrt[3]{27}327​, the radicand is **27**; in 8x2\sqrt{8x^2}8x2​, the radicand is the whole expression **8x28x^28x2**.

In short: the radical sign tells you to take a root, and the radicand is the thing you are taking the root _of_. The radicand can be a plain number, a fraction, a decimal, or an algebraic expression — anything you might want to find the root of. This is the same object you meet whenever you work with [squares and square roots](/content/math/algebra/squares-and-square-roots/index.html); the radicand is just its proper name.

## The Three Parts of a Radical  
A radical expression is built from three pieces. Read them in this order and the whole expression makes sense.

| Part                | What it is                                                        | In 643\sqrt[3]{64}364​     |
|---------------------|-------------------------------------------------------------------|-------------------------------|
| **Radical symbol**  | The x\sqrt{\phantom{x}}x​ sign that signals "take a root"     | x\sqrt{\phantom{x}}x​       |
| **Index**           | The small number at the upper-left giving _which_ root           | 333 (cube root)              |
| **Radicand**       | The number or expression under the symbol                         | 646464                       |

The **index** is the part students most often overlook, because it is usually invisible. When no index is written, it is understood to be **2** — a square root. So 25\sqrt{25}25​ really means 252\sqrt[2]{25}225​; the 2 is just left off by convention. An index of 3 means cube root, an index of 4 means fourth root, and so on. The index must be a whole number of at least 2.

Put together: 643=4\sqrt[3]{64} = 4364​=4 reads as "the cube root (index 3) of the radicand 64 is 4," because 4×4×4=644 \times 4 \times 4 = 644×4×4=64. The whole thing — symbol, index, and radicand together — is called a **radical expression**.

## How Is a Radicand Different From a Radical?  
This is the single most-confused pair in the topic, so it earns a direct answer. The **radical** is the _symbol_ (and, loosely, the whole root expression); the **radicand** is the _content_ under it. One is the operation, the other is the operand.

Think of it like a fraction: the fraction bar is the structure, and the numerator is a part sitting inside that structure. In the same way:

- **Radical** = the root _sign_ x\sqrt{\phantom{x}}x​, the instruction to take a root.
- **Radicand** = what is _underneath_ the sign, the number being rooted.

## What Can a Radicand Be? Positive, Negative, and Fractional  
A common follow-up question: **can a radicand be negative or a fraction?** The honest answer is "it depends on the index," and the distinction is worth getting right.

- **A radicand can be a fraction or a decimal.** 14=12\sqrt{\tfrac{1}{4}} = \tfrac{1}{2}41​​=21​ and 0.25=0.5\sqrt{0.25} = 0.50.25​=0.5 are perfectly ordinary. The radicand here is 14\tfrac{1}{4}41​ or 0.250.250.25.
- **A radicand can be an algebraic expression.** In 2a+5\sqrt{2a + 5}2a+5​, the radicand is 2a+52a + 52a+5 — the whole expression under the sign.
- **A negative radicand depends on the index.** With an _even_ index (square root, fourth root), a negative radicand has **no real value** — no real number squared gives a negative result, so −9\sqrt{-9}−9​ is not a real number. With an _odd_ index (cube root, fifth root), a negative radicand is fine: −273=−3\sqrt[3]{-27} = -33−27​=−3, because (−3)3=−27(-3)^3 = -27(−3)3=−27.

That even/odd split is the seed of complex numbers — the moment a student first meets −1\sqrt{-1}−1​ and learns it is called iii. For now, the rule to carry is simpler: under a square root, keep the radicand at zero or above if you want a real answer.

## Examples of the Radicand  
The six examples move from naming a radicand to reasoning about its sign and structure — the progression a student actually needs.

### Example 1  
**Identify the radicand in 81\sqrt{81}81​.**  
The radicand is whatever sits under the radical sign.  
Radicand=81.\text{Radicand} = 81.Radicand=81.  
**Final answer:** 818181 (and 81=9\sqrt{81} = 981​=9).

### Example 2  
**Identify the radicand and index in 164\sqrt[4]{16}416​ — and avoid the usual slip first.**  
_Wrong attempt._ A student names the radicand as 4 and the index as 16, reading left-to-right and grabbing the first small number they see. Check it against the meaning: the index says _which root_, and a "16th root" of 4 would be a tiny decimal, not the clean value the expression is built for. Something is reversed.  
_Correct._ The small raised number at the upper-left is the **index**, and the number under the symbol is the **radicand**. So the index is **4** and the radicand is **16**.  
164=2since24=16.\sqrt[4]{16} = 2 \quad\text{since}\quad 2^4 = 16.416​=2since24=16.  
The trap is purely positional — the index is small and raised; the radicand sits under the roof. **Final answer:** index 444, radicand 161616.

### Example 3  
**Identify the radicand in 8x23\sqrt[3]{8x^2}38x2​.**  
The radicand is the entire expression under the radical, not just the number.  
Radicand=8x2.\text{Radicand} = 8x^2.Radicand=8x2.  
**Final answer:** 8x28x^28x2 (with index 3, a cube root).

### Example 4  
**Can the radicand be a fraction? Evaluate 916\sqrt{\dfrac{9}{16}}169​​.**  
Yes — the radicand here is 916\tfrac{9}{16}169​. Take the root of the top and bottom separately.  
916=916=34.\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}.169​​=16​9​​=43​.  
**Final answer:** 34\dfrac{3}{4}43​.

### Example 5  
**Is −25\sqrt{-25}−25​ a real number?**  
The radicand is −25-25−25 and the index is 2 (even). An even root of a negative radicand has no real value.  
−25; is not a real number.\sqrt{-25} ; \text{is not a real number.}−25​; is not a real number.  
**Final answer:** undefined in the real numbers (it equals 5i5i5i in the complex numbers).

### Example 6  
**Simplify 72\sqrt{72}72​ by rewriting the radicand.**  
Simplifying a root means breaking the _radicand_ into a perfect-square factor times the rest.  
72=36×2=36,2=62.\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36},\sqrt{2} = 6\sqrt{2}.72​=36×2​=36​,2​=62​.  
The radical sign never moved; only the radicand 72 was rewritten as 36×236 \times 236×2. The same factor-out move is the engine behind [simplifying root 12](/content/math/algebra/square-root-of-12-simplify-root-12/index.html).  
**Final answer:** 626\sqrt{2}62​.

## Why Naming the Radicand Matters  
It is tempting to treat _radicand_ as a vocabulary word to memorise for one quiz and forget. It is more than that — naming the parts of a radical is what lets you _talk about_ and _manipulate_ roots precisely later on.

- **Simplifying radicals** is entirely about rewriting the radicand. "Pull out the perfect-square factor" only makes sense once you can point to the radicand as the thing being factored.
- **The domain of a function.** When a square root appears in an equation, the rule "the radicand must be ≥0\geq 0≥0" is what tells you which inputs are allowed. Solving x−3\sqrt{x - 3}x−3​ starts with "the radicand x−3x - 3x−3 cannot be negative," so x≥3x \geq 3x≥3.
- **The quadratic discriminant.** The b2−4acb^2 - 4ac under the root in the quadratic formula is a radicand — and whether it is positive, zero, or negative tells you everything about a quadratic's roots. That is precisely why the [discriminant](/content/math/algebra/discriminant/index.html) is read as the radicand of the quadratic formula.

## Where Students Trip Up on the Radicand  
The errors here are almost all about _which part is which_ and _what the radicand is allowed to be_. Three come up most.

### Mistake 1: Confusing the radicand with the index  
**Where it slips in:** Reading 83\sqrt[3]{8}38​ and calling 3 the radicand.  
**Don't do this:** Assume the first small number you see is the radicand. The 3 is raised and to the left — that is the index.  
**The correct way:** The radicand is the number _under_ the radical sign; the index is the small number at the upper-left. In 83\sqrt[3]{8}38​, the radicand is **8** and the index is **3**.

### Mistake 2: Treating an even root of a negative radicand as real  
**Where it slips in:** Writing −16=−4\sqrt{-16} = -4−16​=−4.  
**Don't do this:** Assume a negative radicand under a square root gives a negative real answer. Check it: (−4)2=16(-4)^2 = 16(−4)2=16, not −16-16−16, so −4-4−4 cannot be −16\sqrt{-16}−16​.
  
**The correct way:** An even root of a negative radicand has no real value — −16\sqrt{-16}−16​ is undefined in the reals (it is 4i4i4i in the complex numbers). Odd roots are different: −83=−2\sqrt[3]{-8} = -23−8​=−2 is perfectly real.

### Mistake 3: Forgetting the radicand can be a whole expression  
**Where it slips in:** In x+5\sqrt{x + 5}x+5​, naming the radicand as just xxx.  
**Don't do this:** Grab only the first symbol under the sign. Everything under the radical is the radicand.
**The correct way:** The radicand is the entire expression under the sign: x+5x + 5x+5. This matters for domains — the restriction is x+5≥0x + 5 \geq 0x+5≥0, not x≥0x \geq 0x≥0.

## Key Takeaways  
- The **radicand** is the number or expression under a radical sign — in 49\sqrt{49}49​, the radicand is 49.
- A radical has three parts: the radical symbol, the index (which root, defaulting to 2), and the radicand (the value being rooted).
- The radical is the _operation_; the radicand is the _operand_ — don't confuse the two.
- Under an even index a negative radicand has no real value; under an odd index it is fine.
- Naming the radicand is what makes simplifying radicals, finding domains, and reading the discriminant possible.