Radical Formula: Definition, Rules

Radicals, also known as roots are an important concept in Mathematics and Algebra that denote the square root of any number. Radicals can also be used to find the cube root of the number or higher-order roots by figuring out a specific formula that is based on radicals. Radicals are used for simplifying the radical expression and radicals can be seen everywhere around us. The radical symbol is represented by the symbol √. A radical is the opposite of an exponent and any number under a square root symbol is called a radicand. Any number multiplied by itself once is called its square root while multiplying it three times gives it a cube root. Mathematically, we represent cube root as \(3 \sqrt{}\) and square root as √.

Key Terms: Radicals, Radical Formula, Square Root, Cube Root, Algebra, Radical Expressions, Exponent, Equations, Radical Functions, Variable, Radicand, Index 

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What is Radical?

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A radical formula is a mathematical equation that has a radical expression on it. The radical expression can be a cube root, square root or any other root. Radical formulas are simplified by lifting the power of the equation to simplify it on both sides until they match each other. A radical term can be rewritten into a simplified form. The simplified form is where the radicand has all perfect nth roots (square, cube or higher). Radical equation is another name for a radical function. It is an equation in which the unknown is under a root sign. 

The symbol '√' that expresses a root of a number is known as radical and is read as x radical n or nth root of x. The horizontal line that covers the number is called the vinculum and the number under it is called the radicand. The number n written before the radical is called the index or degree.

Parts of a Radical

Solutions of radical equations are those values of the variable which make the equation true. When an equation has a cube or square root with more than one term, an extra step is required to convert the equation into a polynomial form before solving by factoring.

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What is Radical Equation?

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A radical equation is one in which there are roots involved. It can also be defined as an algebraic equation in which there are terms with fractional exponents. Typically, a root or cube root is found on one side of the equation and the rest of the terms are to the other side. A radical equation is an algebraic equation that contains a radical expression. Solving radical equations requires isolating the radical on one side of the equal sign, squaring or cubing both sides of the equation in order to eliminate the radical and solve for the variable.

Radical Equation

Read More: Algebra Formula

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General Rules of Radical Formula

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A radical is an expression that involves the root of a number. Some of the general rules for radicals are given below:

• If a number is positive, then its radical and the result will both be positive. 
• If a number is negative, the same holds true for its radical. 
• The number under the radical will only be considered as irrational when it is negative and any index, even numbers may apply.
• The radical will be square root in case an index is not mentioned.
• Multiplication of numbers under the same radical and index is possible, for instance, \(3 \sqrt{}\)12 × \(3 \sqrt{}\)10 = \(3 \sqrt{}\)120.
• Similarly, division is possible for numbers under the same radical, for instance, √8/√4 = √2.
• When the number is split under the same radical, the reverse of the multiplication rule is possible, for example, √27 = √9 × √3 = \(3 \sqrt{}\)3 × √3.
• Radicals can be written in exponent form as well in any equation, like √x = 25 (√x)2 = (25)2 x = 5.
• The radical is equivalent to the inverse exponent of the index number such as √7 = (7)1/2.

Rules for Radicals

Read More: Integers as Exponents

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Radical Formula

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The radical expression is in the form n√ x. 

x is the radicand, and it should be a variable or an algebraic expression. 

'n' is known as the index of n√ x. 

In order to make an equation of nth root radical free, power both sides of the equation with 'n'. 

n√ x=p

x1/n = p

(x1/n)n = pn

x = pn

This formula helps us to solve radical equations. Using this formula one can change the form of a radical equation-free from radical i.e, from a square root, cube root or higher roots to its simplest form.

Read More: Addition and Subtraction of Algebraic Expressions

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Things to Remember 

• Radicals are represented with a radical sign (√) and a number or expression inside the sign whose root is to be found. There are two parts of a radical - the radicand and the index. 
• The radicand is the value under the radical, whereas the index is always placed on top of the radical and tells us the root to which we need to calculate. All radicals have both, so there cannot be any number on top for example √4.
• Radicals are a mathematical concept that works on the basis of square roots. This guide will focus on some general rules for radicals that will help you to solve radical equations and make life easier.
• Radicals are used to express the root of a number. For example, if you write √25, it means that you want to find the number that when multiplied by itself will give 25. 
• This radical expression is equivalent to writing “the square root of 25”. Since the radical sign is just that, a sign, we could have written the same sentence without the use of radicals and had: “the square root of 25 equals 5”.
• A radical equation is an algebraic equation that contains a radical expression. 

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