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# Radicand|Definition & Meaning

## Definition

The radicand refers to the value or expression within the radical symbol ($\sqrt{}$). Therefore, it is the number whose nth root we want to find. For example, in $\sqrt[2]{a^2+b^2-2ab} = a-b$, the expression $a^2+b^2-2ab$ is the radicand.

In mathematics, the term “**radicand**” refers to the value or expression that is printed **underneath** a **radical** sign or **root** symbol. It is a way of expressing something from which we are drawing inspiration. Take, for instance, the statement which states the **cube root** of **64** is **4**, as an example. In this case, the radicand is the number** 64**.

**The Concept of a Number’s Root**

Both the word “**radicand**” and the word “**radical**” can be traced back to the Latin word “**radix,**” which literally translates to “**root.**” The reasoning for this is that the root, in this case, the root of a word, is the source of whatever it is that it is.

If you take the cube or square of a number, whatever number it was originally derived from becomes the root, whereas the number that was squared or cubed itself becomes the radicand.

These phrases made their first appearance in print in England in the middle of the **1600s** in a book written by **John Pell** entitled **An Introduction to Algebra**.

In **mathematics**, an **expression**, **number**, or **variable** written **inside** a **root** symbol is referred to as a **radicand**. It is a quantity itself for which we are searching for the underlying cause. When we study exponents and roots, we come across the term “**radicand**” quite frequently. Therefore, the value of the radicand **4** is equal to **2**.

The following are some examples of radicands. Here, **x** is the radicand:

**âˆšx**

Here **a + b** is the radicand:

**âˆš(a+b)**

Here is another example where the negative sign is with the square root, so** 9** is a perfect square here; the **negative** sign will come with the answer as it is.

This is the problem that you are attempting to solve by locating the root. The value **(3x)** in the expression**âˆš(3x) **serves as the radicand denoted by the symbol **(3x)**.

An equation is said to be **radical** if it has a minimum of one variable expression inside of a radical, which is often the square root of the variable.

The radical could be any** root**; for example, the **square root** or the **cube root**. In most cases, equations can be solved by separating the variable in question and then undoing whatever changes have been made to that variable.

**Radicand and Radical **

A sign called the **radical **(**âˆš**) is utilized for the purpose of indicating the root of any variable or number. It is written as a symbol. The **radicand** is the word or expression that is placed just under the radical sign. In other words, the radicand is an input to the radical (or nth root) function.

There is one additional concept that pertains to the radicand, and that concept is called the **index**. Let’s get some information on it.

**Index and Radicand **

You must already comprehend the definition of **radicand** by this point in time. Now, if we look closely at the examples that have been presented here, we will notice that the **radicand** sign has a small number printed just of it is often a **natural number** that is bigger than **1**.

For example, in the expression $\mathsf{\sqrt[3]{x+y}}$, the number **3** is placed on the upper left of a **radicand**. In mathematics, this type of quantity is called an **index**. To put it another way, it is the same as the number of the root.

In the event that there is no quantity of index given to a** left** of a **radicand** symbol, we take the index to be **2**, as this is the default.

## When Radicand Is Negative

Principal square roots are obtained by multiplying primary numbers to obtain the **radicand**, and negative values obtained by multiplying primary numbers to obtain the same positive number are referred to as **negative square** roots. If we are given a negative number and asked to find the square of that number, there is no** real number** that can be used.

## When Radicand Is a Perfect Square

If the number that is contained within the radical sign, known as the **radicand**, is a **positive** number and it is possible to factor it as the **squared** of another **positive** integer, therefore the square root of such a number is immediately apparent. The following is a property that we have in this instance:

## When Radicand Is Not a Perfect Square

There is no guarantee that the radicand will always take the form of a** perfect square**. Whenever any **positive** integer does not have a perfect square, then the square root of that **positive integer** will be an irrational number. For instance, here in the figure below, **72** is not a perfect square, so the answer is an irrational number.

Still, you can get a pretty good approximation to the nearest square number, leading to the answer in a **whole number**.

**Example 1**

Solve the following given expression:

**âˆš(64) – 5**

**Solution**

Here the square root is of the perfect square that is radicand is **64,** which is a perfect square of **8:**

** = âˆš(8 ^{2}) – 5**

Here the square will cancel with the square root, so only **8** will be left:

**= 8 – 5**

**= 3**

The final answer to the above expression is** 3**.

**Example 2**

Identify the radicand ** 2 + 2âˆš(121)** and also solve the following expression.

### Solution

The radiant in the given expression is **121. **Here, **121** is a perfect square of **11:**

**121 = 11 ^{2}**

This implies that:

** 2 + 2âˆš(121) = 2 + 2âˆš(11) ^{2}**

**= 2 + 2(11)**

**= 2 + 22**

**= 24**

### Example 3

Solve the following expression evolving radicand **âˆš(3x â€“ 6 ) ^{2}**

**and find the value of**

^{ }**x**.

### Solution

Here radicand is **3x – 6:**

**= âˆš(3x â€“ 6 ) ^{2}**

The square root will cancel with the square:

**âˆš(3x â€“ 6 ) ^{2 }= 3x – 6**

**3x = 6**

**x = 6/3**

**x = 2**

*All the figures above are created on GeoGebra*.