Square and Square Roots
Category : 8th Class
SQUARE & SQUARE ROOTS
FUNDAMENTALS
Square and Square Root
e.g., \[{{\left( 4 \right)}^{2}}=4\times 4=16,{{\left( 5 \right)}^{2}}=5\times 5=25,{{\left( 12 \right)}^{2}}=12\times 12=144\]
Perfect squares or Square number;"
e.g., \[144=2\times 2\times 2\times 2\times 3\times 3={{4}^{2}}\times {{3}^{2}}={{(12)}^{2}}\]
\[81=3\times 3\times 3\times 3={{3}^{2}}\times {{3}^{2}}={{\left( 9 \right)}^{2}}\]
Example:- Show that 300 is not a perfect square.
Solution:- Resolving 300 into prime factors, we get \[300=2\times 2\times 5\times 5\times 3\]
Making pairs of equal factors, we find that the digit 3 is not forming a pair (i.e. it appears only once). Hence 300 is not a perfect square.
Properties of perfect squares
e.g., 82, 73, 177, 2888 etc.
e.g., 160, 4000, 900000 end in one zero, three zero, five zeros. So, none of them is a perfect square.
e.g., \[{{2}^{2}}=4,{{8}^{2}}=64,{{10}^{2}}=100,{{20}^{2}}=576\]etc.
e.g., \[{{(1)}^{2}}=1,{{(9)}^{2}}=81,{{(27)}^{2}}=729\]etc.
e.g., \[{{\left( \frac{2}{3} \right)}^{2}}=\frac{4}{9}\] and \[\frac{4}{9}<\frac{2}{3}\] since \[4\times 3<9\times 2.\]
e.g., \[{{\left( n+1 \right)}^{2}}-{{n}^{2}}=\left( n+1+n \right)\left( n+1-n \right)\]
\[=\left\{ \left( n+1 \right)+n \right\}\]
\[\left\{ {{\left( 36 \right)}^{2}}-{{\left( 35 \right)}^{2}} \right\}=\left( 36+35 \right)\left( 36-35 \right)=71\]
\[\left\{ {{\left( 89 \right)}^{2}}-{{\left( 88 \right)}^{2}} \right\}=\left( 89+88 \right)\left( 89-88 \right)=177\]
e.g., (i) \[1+3+5+7+9={{\left( 5 \right)}^{2}}=25\]
(ii) \[1+3+5+7+9+11+13={{\left( 7 \right)}^{2}}=49\]
e.g., Let n = 1 n+1 = 2
(1)2, (2)2
2, 3 lie between (1)2 and (2)2
\[\Rightarrow \]2n = 2 non-perfect squares numbers between (1)2 and (2)2
Pythagorean Triplets
A triplet (m, n, p) of three natural number, (m, n and p) is called a Pythagoras triplet if \[{{m}^{2}}+{{n}^{2}}={{p}^{2}}.\]
e.g., \[\left( 3,4,5 \right),\left( 5,12,13 \right),\left( 8,15,17 \right)\]etc. are examples of Pythagoras triplets.
Some Shortcuts to find squares
Column Method:- This method is based upon an old Indian method of multiplying two numbers. It is convenient for finding squares of two digit numbers only. This method uses the identity \[{{(x+y)}^{2}}={{x}^{2}}+2xy+{{y}^{2}}\]
e.g., \[{{(52)}^{2}}=\]Let unit place 2 = y and tens place 5 = x
Then follow this Rule
Column-I |
Column-II |
Column-III |
\[{{x}^{2}}\] |
\[2\times x\times y\] |
\[{{y}^{2}}\] |
\[{{(5)}^{2}}\] |
\[2\times 5\times 2\] |
\[{{(2)}^{2}}\] |
\[2\frac{5}{2}\] |
\[2\underline{0}\] |
\[0\underline{4}\] |
27 |
0 |
4 |
Square root:-
The square root of a number y is that number which when multiplied by itself given y as the product. We denote the square root of a number y by\[\sqrt{y}\].
Method to find the square root of a number.
There are two methods:
Follow these steps.
e.g., (1) Find the square root of 576.
Solution:- By prime factorization, we get
\[576=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\]
\[\therefore 576=2\times 2\times 2\times 3=24\]
(2) In an auditorium of Global Pratibha School, the number of rows is equal to the number of chairs in each row. If the capacity of auditorium is 5625, find the number of chairs in each row.
Solution:- Let the number of chairs in each row be x.
Then, the number of rows = x
\[\therefore \]Total number of chairs \[=x\times x={{x}^{2}}\]
\[\therefore \]The capacity = 5625
\[\therefore {{x}^{2}}=5625=5\times 5\times 5\times 5\times 5\times 3\times 3\]
\[\therefore \mathbf{x}=5\times 5\times 3\]
\[\therefore x=75\]
Hence number of chairs in each row = 75.
Square root of the perfect square by the long Division Method
Follow these Steps:-
Take this number as the divisor and also as the quotients
Example-1:- Evaluate \[\sqrt{15876}\]using long division method.
Solution:- Making periods and using the long division method, we have
\[\therefore \sqrt{15876}=126\]
Example-2:- What least number must be added to 5607 to make the sum a perfect square? Find this perfect square and its square roots.
Solution:- Try to find out the square root of 5607.
We observe that
\[{{\left( 74 \right)}^{2}}<5607<{{\left( 75 \right)}^{2}}\]
So, required number to be added
\[{{75}^{2}}-5607=5625-5607=18\]
So, 18 should be added and perfect square = 5625.
Square roots of Number in decimal form
Method makes the number of decimal places even by affixing a zero, if necessary. Now, mark periods and find out the square root by the long division method. Put the decimal point in the square roots as soon as the integral part is exhausted.
Example-3:- Evaluate \[\sqrt{56.25}\]
Solution: -
\[\therefore \sqrt{56.25}=7.5\]
Note:-
(i) \[\sqrt{xy}=\left( \sqrt{x}\times \sqrt{y} \right)\] (ii) \[\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\]
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