SQUARE ROOTS AND CUBE ROOTS math capsule

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I think learning is the most valuable thing we can do to ourselves .

I don't know i may be biased with studies, but i love it to do .

Hope that you will digest this article and start loving the numbers :) 

SQUARE OF A NUMBER

The square of a number is obtained by multiplying it by itself.

e.g. Square of 5 = 5 * 5 = 25 , square of 6 = 6 * 6 = 36

Perfect square or square number -

A natural number n is called a perfect square or a square number , if there exists a natural number m i.e. n = m².

e.g. The numbers 1 , 4 , 9 , 16 , 25 ,........  are square number.

Properties of Square Number 

(1) A number having 2, 3, 7 or 8 at unit's place is never a perfect square.

e.g. 172, 2783

(2) The squares of odd numbers are odd and the squares of even numbers are even.

(3) Square of an even number is even while an odd number is odd.

(4) The difference of squares of two consecutive natural numbers is equal to their sum.

e.g. 7² - 6² = 7 + 6 = 13

(5) The product of two consecutive even or odd natural numbers is (n² - 1), where n is any natural number i.e. (n + 1)(n - 1) = n² - 1

e.g. 13 * 15 = (14 - 1)(14 + 1) = 14² - 1 = 196 - 1 = 195

(6) Every square number is a multiple of 3 or exceeds a multiple of 3 by unity,

e.g.

4² = 16 = 3*5 + 1

5² = 25 = 3*8 + 1

(7) Every square number is a multiple of 4 or exceeds a multiple of 4 by unity.

e.g.             3² = 9 = 4*2 + 1

(8) The perfect square number may ends with the digits 0, 1, 4, 5, 6 and 9.

(9) The product of four consecutive natural numbers plus one is a always square.

(10) A number ending in an odd number of zeroes is never a perfect square.

SQUARE ROOT OF A NUMBER

The square root of  a number x is that number which when multiplied by itself gives x as the product.

In general,

y² = x => y = √x

Here, y is the square root of x, if and only if x is the square of y.

When we take square roots of any number , we will take both positive and negative sign.

The square root of a given number can be determined by using any of the following methods,

1. By prime factorisation

2. By general method/ long division method

SQUARE ROOTS BY PRIME FACTORISATION

In this method we express the given numbers as the product of prime factors . Now, for finding square root , we take the product of these prime factors choosing one out of every pair of same prime factors.

⇨The square root of 213444 is _ a 332      b 368        c 432        d 462

The prime factors of 213444 are ,

√213444 = √2*2*3*3*7*7*11*11 = 2*3*7*11 = 462

Okay now lets see some other method too !

FINDING SQUARE ROOTS BY DIVISION METHOD

Following steps are followed ,

                   STEP 1 

In the given number , place bars over every pair of digits starting with the unit's digit. Each pair and the remaining one digit (if any ) on the extreme left is called a period.

                   STEP 2

Think of the largest number whose square is less than or equal to the first period.

                   STEP 3

Put the quotient above the period and write the product of divisor and quotient just below the first period.

                   STEP 4

Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder . This becomes the new dividend.

                   STEP 5 

Now, the new divisor is obtained by taking two occasions the quotient and annexing with an appropriate digit that is additionally taken because the next digit of the quotient, chosen in such how that the merchandise of the new divisor and this digit is up to or simply but the new  dividend.                

                   STEP 6

Repeat steps two , three and four until all the periods are concerned.

Now , the quotient therefore obtained is that the needed sq. root the given range.

MULTIPLICATION OF RATIONALISATION FACTOR

If a number is given in the form of 1/(√x ± √y) , we multiply by its rationalisation factor (√x ± √y) in both the numerator and denominator.

Done with square roots :)
easy ?
So lets learn about cubes now .

CUBE

If a number is multiplied two times with itself, then the result of this multiplication is called the cube of that number.

e.g. 8 = 2*2*2

CUBE ROOT

The cube root of a given number is the number whose cube is the given number.

In general , we can say that a³ = b, then cube root of 'b' is 'a' . Cube root of a number  'a' is denoted as ∛a .

CUBE ROOTS BY PRIME FACTORISATION

In this method, we express the given numbers as the product of primes factors .

Now , for finding cube root, we take the product of these prime factors choosing one out of every pair of three, from the factors.

⇒ Find the cube root of 373248.

a 42        b 52        c 62          d 72

Here, 373248 = 8*8*8*9*9*9 = 2*2*2 * 3*3*3 * 2*2*2 * 2*2*2 * 3*3*3 = 72

POINTS TO BE REMEMBERED--------------
➔ Cubes of all even numbers are even and all odd numbers are odd.
➜Cubes of the numbers ending in digit 0, 1, 4, 5, 6and 9 are the numbers ending in the same digit.
➜Cubes of negative integers are negative.
➜Cubes of numbers ending in digits 3 and 7 ends in digit 7 and 3 , respectively
➜Cubes of a number which ends in a zero, ends in three zeros.

POWERS OR EXPONENTS

Power of a number 'a' is said to be n, if the result obtained after n times multiplication of a , we get some number b. i.e. nª = b, then we read it as 'n' to the ath power.

e.g 2³ = 8

Here, 2 to the 3th power.

To rate a given number to its indicated power multiply the number by itself as many times as the power indicated.

5⁴ = 5*5*5*5 = 625

Also we sometimes use  5⁴, 4 as exponent and the number 5 is called as base.

LAWS OF EXPONENTS

> (x / y)ª * (x / y)º = (x / y)ª⁺º

> (x / y)ª / (x / y)º = (x / y)ª⁻º

> {(x / y)ª}º = (x / y)ªˣº

> (x / y)⁻ª = (y / x)ª

> (x / y)⁰ = 1

What ?
you read the best blog post today !
thanks :)
Lets do some exercises now.

square roots and cube roots 1

square roots and cube roots 2

square roots and cube roots 3

square roots and cube roots 4

THANK YOU FOR READING THIS AT Math Capsule.