Radicand

A radicand is just the number or expression that sits inside the square root symbol (√). It can be a positive number, a negative number, or an algebraic expression with variables. 

The radical symbol (√) is used to show that you’re finding a square root or another type of root, like cube root. The radicand is the number or expression you’re trying to take the root of. Even when the radical symbol (√) is not shown, understanding the meaning of a radicand helps you easily identify the value from which the root is being taken in an expression.
 

Now it’s your turn to apply what you have learned. Take a look at the following expressions and try to identify the radicand.
 

• What is the radicand in the fourth root expression ∜256?

Answer = 256

The radicand is 256 because it is the number inside the radical symbol, and it is the value we are finding the fourth root of.

The term “radicand” is used in math when working with roots like square roots, cube roots and higher roots. It simply tells us which number or expression is inside the radical and is being used in the operation.
 

This table helps you easily understand the difference between a radical and a radicand. The radical is just the root symbol (like √), and the radicand is whatever is inside it, the number or expression you’re finding the root of.

This table helps you easily tell the difference between a radicand and an index. The radicand is the number or expression inside the root, it’s what you’re working on. The index is the small number placed at the top left of the root symbol (√). It indicates the type of root being taken. The examples clearly show how each part is used in real math problems.

We usually see radicands inside square roots, but they’re not just limited to that. Radicands can also show up in cube roots, fourth roots, or even higher roots.

• Square Root: This is the most common type of root we see. In an expression like √81, the number inside the symbol 81 is called the radicand.

• Cube Root: When a small 3 is written above the root symbol (√), it means you are finding the cube root of the number inside. For example, in ∛64, the radicand is 64.

• Higher Roots: Roots don’t stop at squares or cube roots, you can have fourth, fifth, or even higher roots. The higher the root, the smaller the result tends to be. For example, in ∜16, 16 is the radicand, and we’re taking the fourth root of it.
   

To simplify a radical expression, the first step is to identify the radicand, the number, or expression found inside the square root symbol (√).

For Example,

In √144, the radicand is 144.
In ∛125, the radicand is 125.

Steps to simplify a Radical Expression (With Example):

Example: Simplify √72.

Step 1: Identify the radicand
The radicand is 72

Step 2: Break it into prime factors
 72 = 2 × 2 × 2 × 3 × 3

Step 3: Group the factors
Here we have,

• (2 × 2) a pair of 2s 
• (3 × 3) a pair of 3s
• One 2 left over (no pair here)

Step 4: Move pairs out of the radical

• √(2²) becomes 2

Explanation: 2² = 4
So, √(2²) = √4 = 2
 

• √(3²) becomes 3

Explanation: 3² = 9
So, √(3²) = √9 = 3  
The leftover 2 stays under the root.

So here we get:
√(2² × 3² × 2) = 2 × 3 × √2 = 6√2

So, √72 simplifies to 6√2.
 

When working with square roots, the number inside the root symbol (√) is called the radicand. This helps you simplify expressions correctly. Here are the key rules,

• √ (a²) = a

Taking the square root of a square brings you back to the original number (when it is positive).
Example: √(4²) = √16 = 4

• √(a × b) = √a × √b

We can split a square root over multiplication.
Example: √(9 × 16) = √9 × √16 = 3 × 4 = 12

• √(a / b) = √a / √b (b ≠ 0)

We can also split a square root over division.
Example: √ (25 / 4) = √ 25 / √4 = 5 / 2 = 2.5
 

• √(a + b) ≠ √a + √b

Here, square roots do not work with addition.
Example: √ (9 + 16) = √25 = 5
What should not be done:
√9 + √16 = 3 + 4 = 7. So, √(a + b)  √a + √b

• √(a − b) ≠ √a − √b

They don’t work with subtraction.
Example: √(25 − 9) = √16 = 4
What should not be done:
√25 − √9 = 5 − 3 = 2
 

Here are some helpful tips and tricks to master radicands and to avoid mistakes while practicing problems. 

• Know exactly what a radicand is: Before doing any operations, identify what lies inside the root symbol (√, ∛, etc.). This helps avoid mistakes like confusing index, radical symbol, and radicand.
  
  
   
• Break the radicand into prime-factors: When simplifying something like √72, factor 72 into 36×2, recognize 36 is a perfect square, so √72 = 6√2.
  
  
   
• Realize that the root of a sum is not the sum of roots: For example, √(a + b) ≠ √a + √b. Misapplying this leads to wrong results.
  
  
   
• Check for perfect powers inside the radicand first: Before leaving a radical like √98 as-is, check that 98 = 49×2, so √98 = 7√2. This step makes your simplification stronger and cleaner.
  
  
   
• Watch the denominator: If the radicand appears in a denominator (e.g., 1/√2), rationalize them by multiplying numerator and denominator appropriately, to rewrite without a radical in the denominator.

You might think radicands only live in math books, but they quietly help us out in everyday life too.
 

• In building and designing: When builders or architects need to find the length of a diagonal, like across a room or a ramp, they often use square roots. For example, for figuring out how long a diagonal beam should be they can use the formula √ (length²  + width²).

• Measuring speed and movement: To measure the speed or the movement, square roots help calculate how far or how fast something travels. For example, to find the speed of the roller coaster dropping from the height, square roots are part of the formula.

• In music and sound: Radicands even show up when tuning instruments or adjusting sound waves. They help with frequencies and timing, making your favorite songs sound just right. For example, for adjusting the pitch or the echo in music, we need square root calculations.

• For handling electricity: Electricians and engineers use square roots when they are working with power, voltage, or resistance. For example, if they want to know how strong an electric current is, they might use a formula with a square root.

• Understanding the data: In statistics, square roots help us understand how the data varies, such as the student's score. For example, the standard deviation is found using a square root.