Square root

Square root

If you have to find square of a number then we write

x^2=y

it means we multiplied x two times and got y.

For example find the square of 2 that you have to multiple 2 two times

2^2=2*2=4

We find square of a number by multiplying that number twice.

So what is a square root

Finding a square root is reverse process of square.

In square root we find the number which is multiplied itself to get the square of number.

a square root is represented by symbol

\sqrt{x}

This symbol is known as square root.

here x represents the number whose square root we want to find

In general

if x^2=y then
\sqrt{y}=\sqrt{x*x}=x,
\sqrt{4}=\sqrt{2*2}=2
\sqrt{9}=\squr{3*3}=3

in above example

Square root of y is x
square root of 4 is 2
square root of 9 is 3

Example find the value of \sqrt{36}

We all know 36 is square of 6

so we can write \sqrt{6*6}

so \sqrt{36}=\sqrt{6*6}=6

Find the \sqrt{81}

\sqrt{81}=\sqrt{9*9}=9

Finde the square root of 625

by factoring 625

we get 625=5*5*5*5

so \sqrt{625}

=\sqrt{5*5*5*5}

=5*5

=25

To find the square root of 2 is as below

Square root of 2
Fig: Square Root of 2

Cube root

We can find cube of a number by multiplying it three times

for example to find cube root of x we multiply it tree times

x^3=x*x*x=y

Here y is cube of x

Finding the cube of 2

that is

2^3=2*2*2=8

here 8 is cube of 2

similar

3^3=3*3*3=27

4^3=4*4*4=64

Now what is cube root

Cube root is a reverse process of a cube.

In cube root we find the number which is multiplied itself thrice to get the cube of number.

\sqrt[3]{x}=y here \sqrt[3]{x} is known as cube root of x=y

Cube root is represented by \sqrt[3]{x}


\sqrt[3]{8}=\sqrt{(2*2*2)}=2


\sqrt[3]{x+y}

here x is number whose cube root we want to fine

Finding Square & cube root by factorization

Find th square root of 6084

\sqrt{6084}= \sqrt{2*2*3*3*13*13}=2*3*17=78

Calculate the value of \sqrt{(248+\sqrt{(51+\sqrt{169})})}


\sqrt{(248+\sqrt{(51+\sqrt{169})})}

=\sqrt{(248+\sqrt{(51+13)})} 169 is square of 13

=\sqrt{(248+\sqrt{64})}

=\sqrt{(248+8)}

=\sqrt{256}

=16

HCF and LCM

1) Factors & Multiples– if a number A divides another number B, exactly we say that A is a factor of B. In this case B is called a multiple of A.
2) Highest common factor or a greatest common measure (CGM) or greatest common divisor. The H.C.F of two or more than two numbers is the greatest number that divides each of them exactly.
Method- factorization- express each of the given number as the product of prime factors the product of least power of common prime factors gives H.C.F.
12 = 2×2×3
18 = 2×3×3 H.C.F 2×3 = 6
Division method- to find the H.C.F of two given number divide the larger number by smaller one. now divide the divisor by the reminder, repeat the process of dividing, the proceeding number by the reminder last obtained till zero is obtained as reminder. The last divisor is the required H.C.F same procedure for more than 2 number.
e.g- H.C.F of 513, 1134& 1215
1134 ) 1215 ( 1 H.C.F of 1134 & 1215 = 81
__1134_____
81 ) 1134 ( 14
____1134____
××
H.C.F of 513 & 31

H.C.F of 513 & 81 = 27

Least common multiple (LCM) – the least number which is exactly divisible by each one of the given number is called their L.C.M.
Method- factorization- resource each one of the given numbers into a product of prime factor then L.C.M is the product of highest power so fall the factors.
e.g. 72, 108 & 2100
72 = 2×2×2×3×3 = 2^2×3^2
108 = 2×2×3×3×3 = 2^2×3^3
2100 = 2×2×3×5×5×7 = 2^2×3^1×5^2×7^1
L.C.M = 2^3×3^3×5^2×7 = 37800

Common division method (shortcut method)
Arrange the given number in a row in any order, divide by a number which divides exactly at least two of the given number and carry forwarded the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible. By the same number except the product of the divisor & the co divided numbers is the repaired L.C.M.
e.g- L.C.M of 16, 24, 36, 54

LCM-of-numbers

L.C.M = 2×2×2×3×3×7×3 = 432
4) Product of two numbers = product of their H.C.F & L.C.M
Example 12 × 15 = 3×60 = 180
5) co- primes– the two numbers are said to be co-prime. If their H.C.F is 1
6) H.C.F & L.C.M of fractions
H.C.F = \frac{H.C.F of numerators}{L.C.M of denominators}
Example frac\{2}{3}, frac\{10}{27}, frac\{8}{9}, frac\{16}{81}
H.C.F = H.C.F of 2, 10, 8, 16    =     2
L.C.M of 3, 9,27 81=81

HCF=\frac{2}{81}
L.C.M =\frac{L.C.M of numerators}{H.C.F of denominators}

H.C.F & L.C.M of decimal fraction– the given no make the same number of decimal places by annexing zero in some numbers if necessary considering these numbers without decimal point. Find H.C.F & L.C.M as the case maybe.
e.g. L.C.M & H.C.F of 0.63, 1.05 & 2.10
Without decimal- 63, 105, 210
HCF of 63, 105, 210 is 21
HCF of 0.63,0.105, 2.10 is 0.21
L.C.M of 63, 105, 210 is 630
L.C.M of 0.63, 1.05, 2.10 is 6.30
8) Comparison of Fraction– Find the L.C.M of denominator & convert them equivalent fraction with L.C.M as the denominator by multiples both the numerator & denominator.
Example \frac{1}{2}, \frac{2}{3}=\frac{3}{6}, \frac{4}{6}
then \frac{4}{6} > \frac{3}{6}
Typical Question:- Find the largest number which divided 62, 132, 237 to leave the same remainder in each case.
Reqd. number = H.C.F of (132-62), (237-132), & (237-62)
= H.C.F of 70,105 & 175 = 35