Here you will learn how to find square root and cube root of a number and properties of squares and cubes.
Let’s begin –
Square Root and Cube Root
(a) Square and Square Roots
When any number multiplied by itself, it is called as the square of the number.
Thus, 3 \(\times\) 3 = \(3^2\) = 9
How to find square root of a given number
We will understand this by taking an example. Let 7016 be a given number.
1). Write down the number 7016 as a product of its prime factors,
7016 = \(2 \times 2 \times 2 \times 2 \times 21 \times 21\)
= \(2^4\times 21^2\)
2). The required square root is obtained by having the values of the powers.
Hence. \(sqrt{7016}\) = \(2^2\times 21^1\) = 84
Properties of Squares
1). When a perfect square is written as a product of its prime factors each prime factor will appear an even number of times.
2). The difference between the square of two consecutive natural numbers is always equal to the sum of the natural numbers. Thus, \(41^2\) – \(40^2\) = (40 + 41) = 81
3). The square of a number ending in 1, 5 or 6 also ends in 1, 5 or 6 respectively.
4). The square of any number ending in 5 : The last two digits will always be 25.
5). The square of any number is always non-negative.
Cubes and Cube Roots
When a number s multiplied with itself two times, we get the cube of the number.
Thus, \(x \times x \times x\) = \(x^3\)
How to find cube root of a given number
In order to find the cube root of a number, first we write it in its standard form and divide all powers by 3.
Thus, the cube root of \(3^6 \times 5^9 \times 17^3 \times 2^6\) is given by
\(3^2 \times 5^3 \times 17^ \times 2^2\)
Properties of Cubes
1). When a perfect cube is written in its standard form the values of the powers on each prime factor will be a multiple of 3.
2). The cubes of all numbers (integers and decimals) greater than 1 are greater than the number itself.
3). The value of the cubes of a number between 0 and 1 is lower than the number itself.
4). The cube of a number between 0 and -1 is greater than the number itself.
5). The cube of any number less than -1, is always lower than the number.
