# Roots of Real Numbers and Radicals

Questions with Solutions

Grade 10 questions on roots of numbers and radicals with solutions are presented.

## Definition

*x*is the

*n*root of a number

^{th}*y*is equivalent to

*x*.

^{n}= yFor

*n = 2*, the

*n*root is called the square root .

^{th}For

*n = 3*, the

*n*root is called the cubic root .

^{th}## Examples

1) Since*3*,

^{2}= 9*3*is the square (

*n=2*) root of

*9*.

2) Also since

*(-3)*,

^{2}= 9*-3*is also a square root of

*9*.

3) Since

*(-2)*,

^{3}= -8*-2*is the cubic (n = 3) root of

*-8*

4) Since

*3*and

^{4}= 81*(-3)*, the fourth roots of 81 are 3 and - 3

^{4}=81## Properties of Roots of Real Numbers

1) For*n*even and

*y*positive, there are two

*n*roots of y

^{th}### Example

Since 10^{4}=10000 and (-10)

^{4}= 10000, the fourth roots of 10000 are 10 and -10.

2) For n even and y negative, there are no real n

^{th}roots of y.

### Example

The square root of - 4 is not a real number since no real number x exists such that x^{2}= -4

The fourth root of - 16 is not a real number since no real number x exists such that x

^{4}= -16

3) For n odd, there is always one n

^{th}root of y.

### Example

The cubic (n=3) root of 8 is equal to 2.The fifth root (n=5) of -100000 is equal to -10

## Principal Root

For n even, the principal root is the positive root. For n odd there is only one root and it is the principal root.### Examples

The principal 6^{th}of 64 is equal to 2.

The principal cubic root of -64 is equal to - 4.

## Radical Notation

The symbol √ is called a radical and is used to indicate the principal root of a number as follows:^{n}√y

### Examples

Because of its widespread use, the square root of y is written as √y without indicating the index.

## Questions With Solutions

18.
Use a calculator to approximate the following to 3 decimal places:

Solutions to the Above Problems

## More References and links

Radical Expressions - Questions with Solutions for Grade 10High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers

Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers

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