Rationalizing a Denominator with two terms.
To rationalize a denominator, we need to learn about a conjugate, which is a similar surd but has a different sign. For example, the conjugate of (a+√b) is (a−√b). In mathematics, a number’s conjugate is a number that, when multiplied or added to the given number, yields a rational number. While performing the rationalization, both the numerator and denominator are multiplied with a suitable conjugate.
Example: Rationalize: 1/(4 − √3).
Solution:
Given: 1/(4 − √3)
To rationalize the denominator, multiply and divide the given term with (4 + √3).
= 1/(4 − √3) × (4 + √3)/(4 + √3)
Since, (a + b)(a – b) = a2 – b2
(4 − √3)×(4 + √3) = (4)2 − (√3)2 = 16 − 3 = 13
⇒ 1/(4 − √3) × (4 + √3)/(4 + √3) = (4 + √3)/13
Thus, 1/(4 − √3) = (4 + √3)/13.
Simplify by rationalizing the denominator of (7 + √6)/(3 – √2)
The term “number system” refers to the representation of numbers, where a “number” is a mathematical value used in various mathematical operations such as counting, measuring, labeling, and computation. There are different types of numbers, such as natural numbers, whole numbers, integers, rational and irrational numbers, real numbers, etc. These numbers are used as digits in a number system. Similarly, a number system is classified into various types that have different properties, like a binary number system, an octal number system, a decimal number system, and a hexadecimal number system.
Radicals are an expression with a root, such as a square root, a cube root, a fourth root, etc. If the index of a radical expression is not mentioned, the root is assumed to be a square root. The “n√” is the radical symbol that means “nth root of”. For instance, the “nth root of (a-b)” is symbolically written as shown in the figure given below. Here, “n” refers to the index or degree, “(a-b)” is the radicand, and “(n√)’ is the radical symbol. The root of a whole number with an irrational value is called a surd. For example, √2, 5 + √3, 2√3, etc are some examples of surds.