# 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*^{th} root of a number *y* is equivalent to *x*^{n} = y.

For *n = 2*, the *n*^{th} root is called the square root.

For *n = 3*, the *n*^{th} root is called the cubic root.

## 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*^{4} = 81 and *(-3)*^{4} =81, the fourth roots of 81 are 3 and - 3
## Properties of Roots of Real Numbers

1) For *n* even and *y* positive, there are two *n*^{th} roots of y
### 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
where n is called the index of the radical and y is called the radicand.
### 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 10

High 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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