Current Affairs 8th Class

Category : 8th Class

 Squares and Square Roots

• Square: The square of a number is the product obtained when a number is multiplied by itself.

• Perfect Square: A perfect squares are the shares of whole numbers. Perfect squares are formed by multiplying a whole number by itself.

• Properties of Squares:

(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,

• There are no natural numbers 'm' and 'n' for which \[{{m}^{2}}=2{{n}^{2}}.\]There are 2n non-perfect square numbers between the squares of the numbers n and (n + 1).

• Square root: Square root is the inverse operation of 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

• Determining the square roots of positive numbers without using a calculator

(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.

• Estimating the number of digits in the square root of a given number: Place bars over every two digits from the right. The number of bars obtained is the number of digits in the square root of the number.

• g., \[\sqrt{\overline{9}}=3;\,\sqrt{\overline{25}}=5;\,\,\,\,\sqrt{\overline{100}}=10;\,\,\sqrt{\overline{169}}=13;\,\,\sqrt{\overline{14400}}=120\]

• Estimate the square root of a number by determining the range of the square root of that number.\[\sqrt{193}\]

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.

• To compute the square or square root of a mixed number, first convert it into an improper fraction.