Key Properties and Rules of Radicals
What is a Radical Symbol?
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The radical symbol √ in the subject of mathematics denotes the “root” of that digit.
Radical symbol is employed to signify one of the two inverse operations for exponentiation. Explicitly, it takes the end result of the exponentiation and the exponent used to get it and yields back the base used to get that result with that exponent. In layman's terms, it finds the number that, if multiplied by itself the number of times indicated by the little number by the radical (index), the result will be the number underneath the radical (radicand). If no index is displayed, then the index is 2.
Radicals are generally indicated as fractional (non-integer) powers. You would see \[x^{(\frac{1}{2})}\], \[x^{(\frac{2}{3})}\], \[x^{(\frac{12}{5})}\] as samples. They operate under the simple multiplicative law for exponents, \[(x^{m})\] *\[(x^{n})\] = \[x^{(m+n)}\]. In real numbers, you usually want the Principal Value, so sqrt (4) =2 not -2. You can nest them as well, \[\sqrt{(3 - 2\times \sqrt{(2)}})\] = \[\sqrt{(\sqrt{(2)-1})^{2}})\] = \[\sqrt{(2)}\] - 1
What is a compound radical expression?
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In the above-mentioned radical expression,
“n” is called the index,
“x” is said to be the radicand, and
The math symbol representing the taking of roots is the sign of radical as we already know.
Degree of Radical
The index signifies what root is being taken. If there is no index written, it is understood to be 2, a square root.
When it comes to math, a radical symbol √ is used to represent a radical expression but many people misguidedly read this as a 'square root' symbol, and multiple times it is employed to conclude the square root of a number. However, apart from square roots, it can also be used to denote a cube root, a fourth root, or higher with numbers written in its place accordingly. But it is just in the case of square root that no number is written over the radical symbol.
Now there are a few restrictions on the above-mentioned radical expression that should be kept in mind:
The first factor to keep in mind is that the index, also denoted as “n“ must be a positive integer which is greater than or equal to 2.
The second factor states that the radicand also denoted as “x” must be a real number only.
The third factor being that if the index or “n” is an even integer then the radicand or “x” must be either greater than or equal to the value of zero in order to represent a real number. On the other hand, if the radicand or “x” is an odd number and a negative radicand it will always represent a real number value.
How do we Find the Simplest Radical Form of a Square Root?
In order to find the simplest radical form of a square root, you need to perform prime factorization of the number. To do this, we need to take a number first and start dividing it by various prime numbers till the time all its factors are now prime.
Let us try and find the simplest radical form of number 24.
24, when divided by the smallest divisible prime number 2, gives 12. 12 divided by 2, is 6. 6 divided by 2 is 3.
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Therefore, the prime factorization of 24 is \[2^{3}\] * 3, which in the simplified form means 8 * 3. In the next step, the pair of the same numbers are taken out and kept out of the root symbol. Your placement of the number outside the root symbol should be kept in mind which will otherwise signify that the number placed outside is multiplying the square root of number 24. In this case, there is only one pair in this prime factorization i.e. 2 which will be taken out which leaves us with square root six - time 2 which is the actual answer.
Also know that while we are taking out these pairs, only one digit out of the pair is taken which in this case is number 2 which is kept on the outside of the square root symbol.
Multiplication of Radicals
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Radicals show an easy method of multiplication. For this, all you need to do is multiply the required radical outside the radical sign by the ones inside the symbols.
A radical equation is the one that has minimum one variable expression within a radical, most often the square root.
Solved Examples:
Simplify 2\[\sqrt{4}\]+2\[\sqrt{824}\] + 28.
Simplifying:
Just like with any other expression, when we have a radical expression, we look to form like terms. In this case, like terms are ones that have the same number under the radical signs.
Answer and Explanation:
We can simplify the expression as follows.
\[2\sqrt{4} + 2\sqrt{8}\] = \[2\sqrt{4} + 2\sqrt{2} *4\] = 2 ∗ 2 + 2 ∗ \[2\sqrt{2}\] = \[4 + 4\sqrt{2}\]
∴\[2\sqrt{4} +2\sqrt{8}\] = \[4(1+ \sqrt{2})24 + 28\] = 24 + 22 = 2 ∗ 2 + 2 ∗ 22 = 4 + 42
∴ \[2\sqrt{4} + 2\sqrt{8}\] = 4(1+2).
FAQs on Radicals in Mathematics: Complete Guide for Students
1. What is a radical in mathematics?
In mathematics, a radical is an expression that involves a root of a number, indicated by the radical symbol (√). While it is often used for square roots, it can represent any type of root, such as a cube root or fourth root. For example, √49 is a radical expression that means the square root of 49, which equals 7.
2. What are the main parts of a radical expression?
A radical expression, such as ⁿ√x, consists of three key parts:
- Radical Symbol (√): The symbol used to denote that a root is being taken.
- Radicand (x): The number or variable found inside the radical symbol.
- Index (n): A small number written to the left of the radical symbol that specifies the type of root. For a square root, the index is 2 and is usually not written.
3. How do you simplify a radical expression?
To simplify a radical, you find the largest perfect square (for square roots) or perfect cube (for cube roots) that is a factor of the radicand. For example, to simplify √72:
- Step 1: Find the largest perfect square factor of 72, which is 36.
- Step 2: Rewrite the radical as a product of its factors: √(36 × 2).
- Step 3: Apply the product rule for radicals: √36 × √2.
- Step 4: Simplify the perfect square: 6 × √2.
The simplest form is 6√2.
4. What is the difference between a radical and a square root?
A square root is a specific type of radical where the index is 2. The term radical is a general term that encompasses all types of roots, including square roots, cube roots (index 3), fourth roots (index 4), and so on. Therefore, every square root is a radical, but not every radical is a square root.
5. How are radicals related to fractional exponents?
Radicals are an alternative way to express fractional exponents. An expression with a radical can be converted into an expression with a fractional exponent, which is very useful for solving complex problems. The general rule is ⁿ√x = x1/n.
- For example, a square root like √9 can be written as 91/2.
- Similarly, a cube root like ³√8 can be written as 81/3.
6. Why does the index of a radical matter when the radicand is negative?
The index is crucial because it determines if a real number solution exists for a negative radicand. The rules are:
- Even Index (e.g., square root): If the index is even, the root of a negative number is not a real number. For instance, √-16 is undefined in the real number system.
- Odd Index (e.g., cube root): If the index is odd, the root of a negative number is a real number. For example, ³√-27 = -3, because (-3) × (-3) × (-3) = -27.
7. When is a radical considered to be in its simplest form?
A radical expression is in its simplest form only when all of the following conditions as per the CBSE 2025-26 syllabus are met:
- The radicand contains no factor (other than 1) that is a perfect power of the index.
- The radicand contains no fractions.
- There are no radicals in the denominator of a fraction.
8. Why can't you add or subtract radicals by combining their radicands?
You cannot combine radicands during addition because it violates the fundamental order of operations. Radicals must be treated like variables; you can only combine 'like radicals'—those with the same index and radicand. For example, consider √9 + √16:
- Correct way: First, find the roots. √9 + √16 = 3 + 4 = 7.
- Incorrect way: Adding the radicands first gives √(9 + 16) = √25 = 5.
Since 7 ≠ 5, it shows that radicands cannot be added together directly.
