Surds: Types, Rules, Calculation, Properties & Applications

Surds are expressions that contain a square root, cube root, or another root symbol. Surds are used precisely to write irrational numbers. The irrational numbers' decimals do not terminate or recur, they cannot be written exactly in decimal form. However, such numbers cannot be written as partial fractions too.

Key Takeaways: Surds, indices, multiplication rule, division rule, power rule

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Meaning of Surds

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The Latin word "Surd" means "deaf or mute." Historically, Arabian mathematicians referred to decimal expansion of rational and irrational numbers as audible and inaudible. Because surds are made up of irrational numbers, they were originally known as the same (deaf, dumb) in Arabic and later translated into Latin. √2, √3, and √5 are examples of surds that are not possible to write further in simplified form.

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Types of Surds

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Surds are classified into six different types:

Simple Surd

When the radical formula symbol contains only a number, it is referred to as a simple surd. A monomial or simple surd is one that has only one term. For Example, √5

Pure Surd

Irrational surds are referred to as pure surds. In other words, a surd with no rational factor other than unity is referred to as a pure surd or complete surd. As an example, √3

Similar Surds

Surds that have the same common factorial are referred to as similar surds. For Example, √5, 7√5, 10√5 are similar surds

Mixed Surds

A mixed surd is a number that can be expressed as a product of rational and irrational numbers. If a portion of the quantity under the radical sign is subtracted from it, the mixed surd is formed.

Compound Surds

A complex surd is formed by adding or subtracting two or more surds. It is the algebraic sum of two or more simple surds or the algebraic sum of a rational number and simple surds.

Binomial Surd

When the sum of two quadratic surds or a quadratic surd and a rational number is multiplied by the difference of those two quadratic surds or a quadratic surd and a rational number, the rational number under the root of surd is squared off and it becomes a rational number because the product of the sum and difference of two numbers is the difference of the square of the two numbers.

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Rules of Surds

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Following are the rules of Surds:

Rule 1

\(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\)

For Example:

Simplify \(\sqrt{18}\)

Since 18 = 9 × 2 = 3² × 2, 9 is the largest perfect square factor of 18.

\(\therefore \sqrt{18} = \sqrt{3^2 \times 2}\)

\( = \sqrt{3^2} \times \sqrt{2} \)    (using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\))

\(=3\sqrt{2}\)

Rule 2

\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

For Example:

Simplify \(\sqrt{\frac{12}{121}}\)

\(\sqrt{\frac{12}{121}} = \frac{\sqrt{12}}{\sqrt{121}}\)    (using the rule \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\))

\(= \frac{\sqrt{2^2 \times 3}}{11}\)        (since 4 is the largest perfect square factor of 12)

\(= \frac{\sqrt{2^2} \times \sqrt{3}}{11}\)        (using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\))

\(= \frac{2\sqrt{3}}{11}\)

Rule 3

\(\frac{b}{\sqrt{a}} = \frac{b}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{b\sqrt{a}}{a}\)

By multiplying both the numerator and denominator by the denominator you can rationalize the denominator.

For Example:

Rationalise \(\frac{5}{\sqrt{7}}\)

\(\frac{5}{\sqrt{7}} = \frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}\)    (multiply both numerator and denominator by \(\sqrt{7}\))

\(= \frac{5\sqrt{7}}{7}\)

Rule 4

\(a\sqrt{c}\ \pm \ b\sqrt{c} = (a\pm b)\sqrt{c}\)

For Example:

Simplify \(5\sqrt{6} + 4\sqrt{6}\)

\(5\sqrt{6} + 4\sqrt{6} = (5+4) \sqrt{6}\)            (using the rule \(a\sqrt{c}\ \pm \ b\sqrt{c} = (a\pm b)\sqrt{c}\) )

\(= 9 \sqrt{6}\)

 Rule 5

\(\frac{c}{a+b\sqrt{n}}\)  multiply top and bottom by  \(a - b\sqrt{n}\)

Following this rule enables you to rationalise the denominator.

For Example:

Rationalise \(\frac{3}{2+\sqrt{2}}\)

\(\frac{3}{2+\sqrt{2}} = \frac{3}{2+\sqrt{2}} \times \frac{2-\sqrt{2}}{2-\sqrt{2}}\)             (multiply the numerator and denominator by \(2-\sqrt{2} \))

\(=\frac{6-3\sqrt{2}}{4-2}\)

\(=\frac{6-3\sqrt{2}}{2}\)

Rule 6

\(\frac{c}{a-b\sqrt{n}}\)  multiply top and bottom by  \(a + b\sqrt{n}\)

Following this rule helps to rationalise the denominator.

For Example:

Rationalise \(\frac{3}{2-\sqrt{2}}\)

\(\frac{3}{2-\sqrt{2}} = \frac{3}{2-\sqrt{2}} \times \frac{2+\sqrt{2}}{2+\sqrt{2}}\)             (multiply the numerator and denominator by \(2+\sqrt{2} \))

\(=\frac{6+3\sqrt{2}}{4-2}\)

\(=\frac{6+3\sqrt{2}}{2}\)

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Calculation of Surds

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Surds must be simplified before calculations can be performed. Surd simplification is accomplished in two simple steps.

STEP 1: Divide the number contained within the root into its prime factors.

STEP 2: Write the prime factors based on the root outside the root. Write one factor outside the root for every two similar factors within the root in the case of square roots.

Example:

Calculate √18 + √50

\(\sqrt{18} + \sqrt{50} = \sqrt{(3 \times 3 \times 2)} + \sqrt{(5 \times 5 \times 2)}\)

\(= 3\sqrt{2} + 5\sqrt{2}\)

\(= 8 \sqrt{2}\)

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Properties of Indices

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Following are the Properties of Indices:

Multiplication Rule

1.  \(x^n \times x^m = x^{m+n}\)
2.  \(x^n \times y^n = (x \times y)^n\)

Division Rule

1.  \(\frac{x^n}{y^n}= (\frac{x}{y})^n\)
2.  \(\frac{x^n}{x^m}= x^{n-m}\)

Power Rule

1.  \((x^n)^m = x^{m \times n}\)
2.  \(x^{n^m} = x^{(n^m)}\)
3.  \(\sqrt[m]{x^n} = x^{\frac{n}{m}}\)
4.  \(\sqrt[m]{x} = x^{\frac{1}{m}}\)
5.  \(x^{-n} = \frac{1}{x^n}\)

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Application of Surds

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Following are the Application of Surds:

• Surds and Indices are used to ensure the accuracy of important calculations.
• Indices are used in a variety of disciplines, including computer games, physics, pH, accounting, finance, and many others.
• Exponential growth is a critical aspect of finance, demography, biology, economics, resources, electronics, and many other fields, and indices are widely used in these and other fields.

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Points to Remember

Following are some important points:

• Surds are representations of numbers in the form of root mean square because these numbers cannot be whole or rational. These numbers cannot be expressed as fractions.
• Surds is a simple irrational number under square root, but it becomes more complicated when two or more surds are combined or when surds are linked to topics such as integration and trigonometry.
• Surd is a positive real number under the square root. Surds provide a platform to use algebra knowledge to solve sums, and its theories and rules help to solve complex trigonometry and integration.
• If the denominator of a fraction has any surds, rationalise it by multiplying both the numerator and the denominator by a conjugate surd.
• Gherardo of Cremona appears to have been the first European mathematician to use the surds terminology.

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