Rationalization

Rationalization is the process of making a complex fraction more manageable. This is achieved by rewriting the expression in a simpler, more standard form. The idea is to make the numbers easier to work with while retaining their values. 

Radical

An expression that uses a root, such as a square root or cube root, is called a radical. For example, √(a + b) is a radical expression of (a + b).

Radicand

The radicand is a number or expression written inside the radical symbol. For example, 3a + b: here, a + b is the radicand and we need to find its cube root.

Radical Symbol

The symbol √ is read as “the root of.” If the radicand is √16, then it is read as “the square root of 16.”  The horizontal bar over the radicand is called the vinculum. It represents variables or constants in the root function. If the variables or constants do not appear beneath the vinculum, then it means they are not part of the root. 

Degree

The degree denotes the root type: 2 for square roots, 3 for cube roots, and so on. If the degree is not mentioned, then it is understood that the radical symbol is a square root.

Conjugate

A conjugate is a binomial with the same terms but the opposite sign. For example, the conjugate of the expression x + y is x - y and vice versa.
 

Surds are irrational numbers in root form. They are left in root form because they cannot be written as a simple fraction. For example, √2 is a surd because its decimal values are non-repeating and non-terminating. Hence, it cannot be expressed as a simple fraction. The method used for rationalizing these expressions depends on whether the radical is part of a monomial or a binomial expression:

Examples for monomial radical: √5, 2√7, 5√3, √11, 4√10.

Examples for binomial radicals: √2 + 3, 2 - √5, 1 + 2√6, 5 - √8.

How to Rationalize a Monomial Radical?

A monomial has a single term, such as √2, √7x, or 3√7x. For expressions with simple radicals as their denominators, we can simply multiply both the numerator and the denominator by the denominator. For a monomial like a ÷ ⁿ√xⁿ (where n < m), multiply by ⁿ√x(m−n) to make the denominator a perfect root, which can then be simplified. For example, let us rationalize the fraction: 53.

Step 1: Evaluate the fraction - the provided fraction 53 has √3 in the denominator, which is a monomial radical. It is important to note that the numerator can contain a radical, thus, you should not be concerned about it while evaluating or simplifying the fraction. 

Step 2: Multiply the numerator and denominator by √3 (the radical in the denominator).

             53 × 33 = 533

Step 3: The final answer is  533.
 

If the denominator has a binomial with a radical like a + b, multiply both the numerator and the denominator by its conjugate (a - b). For example, let us rationalize the fraction 53 + 5.

Step 1: We want to remove the radical from the denominator. Since the denominator is a binomial with a surd, we multiply both the numerator and denominator by its conjugate.

Step 2: Multiply the fraction by the conjugate of the denominator, which is 3 − √5:
             53 + 5 × 3 - 53 - 5

Step 3: Use the identity: (a+b) (a−b) = a2− b2.
             So the denominator becomes: (3 +5​) (3 −5​) = 9 − 5 = 4

Step 4: Now multiply the numerators: 5​(3−5) = 35−5

Step 5: Put it all together, 3 5 - 5 4 and the final answer is 3 5 - 5 4.

Let us look at this with an example for better understanding. Let us rationalize 134:

Step 1: Examine the fraction. The denominator contains a radical in the form of a cube root.

Step 2: We need to make 34 a whole number because we don’t want a cube root in the denominator. To achieve this, we need to multiply 34  by something so that we get a perfect cube as the product. Now, the number that we are looking for is 364  because 364  = 4. So we need to multiply the denominator by 316  because 34   316     = 364  = 4

Step 3: Multiply the numerator and the denominator by 316 
 
 134  × 316316 = 316364 = 3164.
 

We need to rationalize a denominator to make our calculations easy. Let us take a look at the step by step process of rationalizing a denominator:

Step 1: Identify the radical in the denominator and proceed accordingly. For e.g., if the numerator is 1 and the denominator is 3, we need to remove the square root in the denominator. To do so, we can multiply both the numerator and denominator by 3.

Step 2: Do the simplification of surds. Make sure any square roots are simplified as much as possible. For example, √8 should be written as 2√2 because that’s its simplest form.

Step 3: At last, simplify the entire fraction. After rationalizing, you may be able to reduce the expression by dividing or combining like terms. The final answer should be in simplified form.
 

Understanding the value of rationalization allows us to simplify phrases and solve issues more efficiently. Here are some real-world examples of rationalization:

• Engineering and Construction: Rationalization simplifies measurements involving square roots, such as lengths, areas, or diagonal distances. This is important because simplifying complex numbers makes it easier to execute precise architectural plans.

• Finance and Data Analysis: In financial modeling or statistical formulas, rationalizing expressions reduces unneeded complexity, resulting in clearer insights and faster calculations.

• Computer Graphics and Animation: In graphics programming, rationalizing complex mathematical equations boosts performance and lowers computational mistakes, particularly when dealing with geometric transformations or 3D modeling.