Squares and Square roots
Category : 8th Class
Squares and Square Roots
(i) A number ending in 2, 3, 7 or 8 is never a perfect square. All square numbers end in 0, 1,4,5,6 or 9.
(ii) A number ending in an odd number of zeroes is never a perfect square.
(iii) Square numbers have only even number of zeros at the end.
(iv) Squares of even numbers are even.
(v) Squares of odd numbers are odd.
(vi) For every natural number
\[n,{{\left( n+1 \right)}^{2}}\text{ }-{{n}^{2}}=\left( n+1 \right)+\text{ }n.\]
e.g.,\[{{9}^{2}}-{{8}^{2}}=9+8=17\]A triplet (a, b, c) of three natural numbers 'a; 'b' and 'c' is called a Pythagorean triplet
If \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\]
(viii) For any natural number m > 1, we have \[{{(2m)}^{2}}+{{({{m}^{2}}-1)}^{2}}=({{m}^{2}}+So,2m,({{m}^{2}}-1)\]and \[\left( {{m}^{2}}\text{ }+\text{ }1 \right)\]form a Pythagorean triplet.
(ix) The square of a natural number 'n' is equal to the sum of the first 'n' odd numbers.
(x) If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square,
(i) The square root of a number \['x'\] is a number which when multiplied by itself gives\['x'\] as the product. We denote the square root of \['x'\] by \[\sqrt{x}\]
(ii) There are two integral square roots of a perfect square number. The positive square root of a number is denoted by the symbol \[\sqrt{{}}\]
(iii) If x and y are positive numbers, work out the square root of the numerator and denominator separately.\[\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\]
(vi) Square root of a number can be found using the following methods.
(a) Repeated subtraction (b) Prime factorisation and (c) Division
(i) The square root of a fraction is determined by finding the square root of the numerator and denominator separately.
(ii) Some fractions must be reduced to fractions with perfect squares as their numerators and denominators before their square roots can be calculated.
(iii) To find the square root of a mixed number, first change the mixed number into an improper fraction.
(iv) The square root certain decimals are obtained by first changing the decimals into fractions with perfect squares as their numerators and denominators.
169 < 193 < 196 \[\leftarrow \] Determine the range between two known perfect squares.
\[\sqrt{169}\,<\sqrt{193}<\sqrt{196}\,\leftarrow \]Square root the range.
\[13\,<\sqrt{193}\,<14\leftarrow \]Estimated answer.
(i) The square root of a number with one bar has one digit.
(ii) The square root of a number with two bars has two digits. The square root of a number with three bars has three digits.
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