Square Root and Cube Root of Surds: Definition, Calculator, Questions, Formula

What are Surds?

Surds are irrational numbers represented in root form. They cannot be simplified further into rational numbers and are commonly encountered in algebra, geometry, trigonometry, and higher mathematics. Understanding surds is essential for solving square root and cube root problems in school mathematics and competitive examinations.

Surds Meaning in Simple Words

A surd is a root expression whose value is irrational.

In simple words, if the square root, cube root, or higher root of a number cannot be expressed as an exact rational number, it is called a surd.

Examples

$\sqrt{2}$

$\sqrt{3}$

$\sqrt{5}$

$\sqrt[3]{7}$

These numbers cannot be written as simple fractions and have non-terminating, non-repeating decimal values.

Definition of Surds

A surd is an irrational root expression that cannot be simplified into a rational number.

Mathematically:

$\sqrt[n]{a}$

is a surd if $a$ is not a perfect $n^{th}$ power.

Examples

$\sqrt{2}$

$\sqrt{11}$

$\sqrt[3]{5}$

Non-Examples

$\sqrt{25}=5$

$\sqrt{81}=9$

$\sqrt[3]{64}=4$

Since these simplify to rational numbers, they are not surds.

Real-Life Examples of Surds

Surds frequently appear in practical situations involving measurements and geometry.

Examples

Example

For a square of side 1 unit:

$\text{Diagonal}=\sqrt{1^2+1^2}$

$=\sqrt{2}$

Thus, the diagonal length is a surd.

Why Surds are Important in Mathematics

Surds are important because they:

• help represent irrational numbers exactly

• simplify algebraic calculations

• are used in coordinate geometry

• appear in trigonometry and mensuration

• help solve quadratic equations

• are frequently asked in aptitude and entrance examinations

Applications of Surds

Structure of Surds

Every surd consists of specific components that determine its value and form.

Understanding the structure of surds helps in simplification and arithmetic operations.

Components of a Surd

A surd generally contains:

• Radical sign

• Radicand

• Index (in some cases)

Example

$\sqrt[3]{54}$

Here:

• Radical sign = $\sqrt{\phantom{x}}$

• Radicand = 54

• Index = 3

Radical Sign, Radicand, and Index

The different parts of a surd are defined below.

Example

In:

$\sqrt[4]{81}$

• Radical sign = $\sqrt{\phantom{x}}$

• Radicand = 81

• Index = 4

For square roots, the index is generally omitted.

Example:

$\sqrt{16}$

Here, the index is 2.

Standard Form of a Surd

A surd is said to be in standard form when:

• The radicand contains no perfect square factors (for square roots).

• The radicand contains no perfect cube factors (for cube roots).

Examples

$\sqrt{12}=2\sqrt{3}$

$\sqrt{50}=5\sqrt{2}$

$\sqrt[3]{54}=3\sqrt[3]{2}$

These are standard forms because the surd part cannot be simplified further.

Rational and Irrational Parts of a Surd

A surd may contain both rational and irrational components.

Example

$5\sqrt{2}$

Here:

• Rational part = 5

• Irrational part = $\sqrt{2}$

Similarly,

$3\sqrt[3]{7}$

contains:

• Rational part = 3

• Irrational part = $\sqrt[3]{7}$

Types of Surds

Surds are classified into different categories based on their structure and characteristics.

Simple Surds

A simple surd contains only one irrational root.

Examples

$\sqrt{2}$

$\sqrt{7}$

$\sqrt[3]{5}$

These surds contain a single radical expression.

Compound Surds

A compound surd contains two or more surds combined through addition or subtraction.

Examples

$\sqrt{2}+\sqrt{3}$

$\sqrt{5}-\sqrt{2}$

$\sqrt{7}+\sqrt{11}$

Compound surds are commonly encountered in algebraic simplification problems.

Pure Surds

A pure surd has no rational factor outside the radical sign.

Examples

$\sqrt{2}$

$\sqrt{11}$

$\sqrt[3]{7}$

Only the irrational root is present.

Mixed Surds

A mixed surd contains both a rational coefficient and an irrational root.

Examples

$2\sqrt{3}$

$5\sqrt{2}$

$7\sqrt[3]{4}$

These are obtained after simplifying surds.

Similar and Dissimilar Surds

Similar Surds

Surds having the same irrational part are called similar surds.

Examples:

$2\sqrt{5}$

$7\sqrt{5}$

$10\sqrt{5}$

All contain the same surd part $\sqrt{5}$.

Dissimilar Surds

Surds having different irrational parts are called dissimilar surds.

Examples:

$\sqrt{2}$

$\sqrt{3}$

$\sqrt{5}$

These cannot be directly combined through addition or subtraction.

Square Roots of Surds

Square roots of surds are obtained by finding the square root of an irrational root expression.

They are commonly used in algebra and advanced simplification problems.

Meaning of Square Root of a Surd

The square root of a surd is another surd whose square equals the original surd.

Example

$\sqrt{\sqrt{16}}$

$=\sqrt{4}$

$=2$

For irrational surds:

$\sqrt{\sqrt{2}}$

$=2^{1/4}$

which is also a surd.

How to Find the Square Root of a Surd

The easiest method is to express the surd using exponents.

Formula

$\sqrt{\sqrt[n]{a}}=a^{\frac{1}{2n}}$

Example

$\sqrt{\sqrt{3}}$

$=(3^{1/2})^{1/2}$

$=3^{1/4}$

Simplifying Square Roots of Surds

Use exponent rules and prime factorization.

Example 1

$\sqrt{12}$

$=\sqrt{4\times3}$

$=2\sqrt{3}$

Example 2

$\sqrt{75}$

$=\sqrt{25\times3}$

$=5\sqrt{3}$

Solved Examples of Square Roots of Surds

Example 1

Find:

$\sqrt{50}$

Solution:

$\sqrt{50}$

$=\sqrt{25\times2}$

$=5\sqrt{2}$

Example 2

Find:

$\sqrt{108}$

Solution:

$\sqrt{108}$

$=\sqrt{36\times3}$

$=6\sqrt{3}$

Cube Roots of Surds

Cube roots of surds involve irrational root expressions whose cube roots cannot be expressed as rational numbers.

These concepts are important in algebra, indices, and higher mathematics.

Meaning of Cube Root of a Surd

The cube root of a surd is a number which, when cubed, gives the original surd.

Example

$\sqrt[3]{\sqrt{2}}$

$=(2^{1/2})^{1/3}$

$=2^{1/6}$

which is also a surd.

How to Find the Cube Root of a Surd

Convert the radical expression into fractional exponent form.

Formula

$\sqrt[3]{\sqrt[n]{a}}=a^{\frac{1}{3n}}$

Example

$\sqrt[3]{\sqrt{5}}$

$=(5^{1/2})^{1/3}$

$=5^{1/6}$

Simplifying Cube Roots of Surds

Extract perfect cube factors from the radicand.

Example 1

$\sqrt[3]{54}$

$=\sqrt[3]{27\times2}$

$=3\sqrt[3]{2}$

Example 2

$\sqrt[3]{128}$

$=\sqrt[3]{64\times2}$

$=4\sqrt[3]{2}$

Solved Examples of Cube Roots of Surds

Example 1

Find:

$\sqrt[3]{250}$

Solution:

$\sqrt[3]{250}$

$=\sqrt[3]{125\times2}$

$=5\sqrt[3]{2}$

Example 2

Find:

$\sqrt[3]{686}$

Solution:

$\sqrt[3]{686}$

$=\sqrt[3]{343\times2}$

$=7\sqrt[3]{2}$

These square roots and cube roots of surds form the foundation for advanced topics such as rationalization, indices, algebraic simplification, and higher mathematics.

Square roots and Cube roots: An Overview

Square roots and cube roots are fundamental operations in mathematics. The square root of a number $x$ is a value $y$ such that $y^2 = x$, and the cube root of $x$ is a value $y$ such that $y^3 = x$. These operations are crucial for solving equations, simplifying expressions, and understanding various mathematical concepts.

How to find the Square root of a number

Finding the square root of a number is a fundamental mathematical operation that involves determining a number that, when multiplied by itself, yields the original number. Various methods such as long division, factorisation, and approximation are used to compute square roots accurately.

Long Division Method

The long division method for finding the square root of a number involves a step-by-step process of division and estimation. Here’s how it works:

• Step 1: Group the number into pairs of digits starting from right to left.

• Step 2: Starting with the leftmost pair, find the largest integer whose square is less than or equal to the first pair of digits. This forms the first part of the square root.

• Step 3: Subtract the square of this integer from the original pair and bring down the next pair of digits to form a new dividend.

• Step 4: Double the current part of the square root found so far and find the next digit that, when multiplied by itself, gives a number close to the new dividend.

• Step 5: Repeat steps 3 and 4 until all pairs of digits have been used, forming the complete square root.

Example: Find the square root of 1225 using the long division method.

Method of factor

The factorisation method for finding the square root of a number involves the following steps:

• Step 1: Factorize the number into its prime factors.

• Step 2: Take one factor from each pair of these prime factors.

• Step 3: Multiply these to obtain the square root.

Example: Find the square root of 324 using the factorisation method.

So, the square root of 324 is (2 × 3 × 3) = 18.

Method of Approximation

This method involves estimating the square root by identifying two close perfect squares and interpolating them.

The square root of a Surd

The square root of a surd refers to simplifying an expression that contains a square root of a number that is not a perfect square. Surds often involve radicals (square roots) of numbers that cannot be simplified further into rational numbers. Finding the square root of a surd typically involves using algebraic techniques to express it in a simplified form.

Steps to find the square root of a surd

Let’s find the square root of the surd ($a+\sqrt{b}$) following these general steps:

• Use the formula, $(a + b)^2 = a^2 + 2ab + b^2$, to simplify the expression of $\sqrt{a + \sqrt{b}}$.

• Identify $x$ and $y$ such that $\sqrt{a + \sqrt{b}} = (\sqrt{x} + \sqrt{y})^2$.

• The square root of ($a+\sqrt{b}$) will be $(\sqrt{x} + \sqrt{y})$.

Example: Find the square root of $(9+4\sqrt{5})$.

Here, $9+4\sqrt{5}$ = $4+5+4\sqrt{5}$ = $(2)^2+2×2×\sqrt{5}+(\sqrt{5})^2$ = $(2+\sqrt{5})^2$

Hence, the square root of $(9+4\sqrt{5})$ is $(2+\sqrt{5})$ or $-(2+\sqrt{5})$.

How to find the Cube root of a number

Finding the cube root of a number involves determining a value which, when multiplied by itself three times, equals the original number. There are various ways to find the cube root of a number, let's discuss them with proper examples.

Method of factor

This method is similar to finding the square root by factorisation but involves triplets of factors. Finding the cube root of a number using the factorization method involves the following steps:

• Break down the number into its prime factors.

• Group the prime factors into sets of three identical factors.

• For each group of three identical factors, take one factor out of the group.

• Multiply these factors together to get the cube root of the original number.

Example: Find the cube root of 1728 using the factorisation method.

Method of Approximation

Estimate the cube root by identifying two close perfect cubes.

Example: Find the cube root of 17 using the approximation method.

Cube root of a Surd

The cube root of a surd involves simplifying an expression that includes a cube root of a number that is not a perfect cube. Finding the cube root of a surd generally requires using algebraic methods such as using specific formulas to simplify and express the cube root in its simplest form.

Steps to find the cube root of a surd

Let’s find the cube root of the surd ($a+\sqrt{b}$) following these general steps:

• Use the formula, $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$, to simplify expression of $\sqrt{a + \sqrt{b}}$.

• Identify $x$ and $y$ such that $\sqrt{a + \sqrt{b}} = (\sqrt{x} + \sqrt{y})^3$.

• The cube root of ($a+\sqrt{b}$) will be $(\sqrt{x} + \sqrt{y})$.

Example: Find the cube root of $10 + 6\sqrt{3}$.

Here, $10 + 6\sqrt{3}$

= $1 + 9 + 3\sqrt{3} + 3\sqrt{3}$

= $(1)^3 + 3×(1)^2×(\sqrt{3}) + 3×(1)×(\sqrt{3})^2 + (\sqrt{3})^3$

= $(1+\sqrt{3})^3$

So, the cube root of $10 + 6\sqrt{3}$ is $(1+\sqrt{3})$.

Best Books for Square Roots, Cube Roots, and Surds

A strong understanding of surds, square roots, and cube roots is essential for mastering number systems, algebra, simplification, and quantitative aptitude. The following books are highly recommended for both concept building and exam preparation.

Shortcut Tips and Tricks for Square Roots and Cube Roots of Surds

Surd questions can often be solved quickly by identifying perfect square and perfect cube factors. These shortcuts are particularly useful in competitive examinations.

Tips to Solve Surds Questions Quickly

These practical tips can improve speed and accuracy in school and competitive examinations.

Important Formula Table for Square Roots and Cube Roots of Surds

The formulas below are frequently used in surds, indices, algebra, and quantitative aptitude.

Important Perfect Squares for Surds

Important Perfect Cubes for Surds

These tables provide a quick revision sheet for surds and indices, square roots of surds, cube roots of surds, irrational numbers, algebraic simplification, and quantitative aptitude preparation.

Practice Questions/Solved Examples based on square and cube roots of surds