Class 8: Square Roots & Cube Roots (Lecture Notes)

$\displaystyle \textbf{Square of a Number}$
$\displaystyle \text{The square of a number is that number raised to the power }2.$
$\displaystyle \text{Example: Square of }9=9^2=9\times9=81.$
$\displaystyle \text{Square of }0.2=(0.2)^2=0.2\times0.2=0.04.$
$\displaystyle \\$

$\displaystyle \textbf{Perfect Square}$
$\displaystyle \text{A natural number is called a perfect square if it is the square of some}$
$\displaystyle \text{natural number.}$
$\displaystyle \text{Example: }1^2=1,\qquad2^2=4,\qquad3^2=9.$
$\displaystyle \therefore 1,\ 4\text{ and }9\text{ are perfect squares.}$
$\displaystyle \\$

$\displaystyle \textbf{Some Properties of Squares of Numbers}$
$\displaystyle \text{(i) The square of an even number is always an even number.}$
$\displaystyle \text{Example: }2\text{ is even and }2^2=4,\text{ which is even.}$

$\displaystyle \text{(ii) The square of an odd number is always an odd number.}$
$\displaystyle \text{Example: }3\text{ is odd and }3^2=9,\text{ which is odd.}$

$\displaystyle \text{(iii) The square of a proper fraction is a proper fraction less than the}$
$\displaystyle \text{given fraction.}$
$\displaystyle \text{Example: }\left(\frac{1}{2}\right)^2 =\frac{1}{2}\times\frac{1}{2}=\frac{1}{4},\quad\frac{1}{4}<\frac{1}{2}.$

$\displaystyle \text{(iv) The square of a decimal number between }0\text{ and }1\text{ is smaller}$
$\displaystyle \text{than the given decimal number.}$
$\displaystyle \text{Example: }0.1<1\text{ and }(0.1)^2=0.1\times0.1=0.01<0.1.$

$\displaystyle \text{(v) A number ending in }2,\ 3,\ 7\text{ or }8\text{ is never a perfect square.}$
$\displaystyle \text{Example: The numbers }72,\ 243,\ 567\text{ and }1098\text{ end in }2,\ 3,\ 7$
$\displaystyle \text{and }8\text{ respectively. Hence, none of them is a perfect square.}$

$\displaystyle \text{(vi) A number ending in an odd number of zeros is never a perfect square.}$
$\displaystyle \text{Example: The numbers }690,\ 87000\text{ and }4900000\text{ end in one, three}$
$\displaystyle \text{and five zeros respectively. Hence, none of them is a perfect square.}$
$\displaystyle \\$

$\displaystyle \textbf{Square Root}$
$\displaystyle \text{The square roots of a positive number }x\text{ are the numbers which, when}$
$\displaystyle \text{multiplied by themselves, give }x\text{ as the product.}$
$\displaystyle \text{The symbol }\sqrt{x}\text{ denotes the positive (principal) square root of }x.$
$\displaystyle \text{Example: Since }7^2=49\text{ and }(-7)^2=49,\text{ the square roots of }49$
$\displaystyle \text{are }7\text{ and }-7.\text{ However, }\sqrt{49}=7.$
$\displaystyle \\$

$\displaystyle \textbf{Methods of Finding the Square Root of Numbers}$
$\displaystyle \textbf{Square Root of a Perfect Square Using Prime Factorization}$
$\displaystyle \text{1. Resolve the given number into its prime factors.}$
$\displaystyle \text{2. Make pairs of equal prime factors.}$
$\displaystyle \text{3. Take one prime factor from each pair and find their product.}$
$\displaystyle \text{The product so obtained is the square root of the given number.}$
$\displaystyle \\$

$\displaystyle \textbf{Test for a Number to be a Perfect Square}$
$\displaystyle \text{A given number is a perfect square if, in its prime factorization, every}$
$\displaystyle \text{prime factor occurs an even number of times.}$
$\displaystyle \text{In other words, its prime factors can be grouped into pairs of equal factors.}$
$\displaystyle \text{Example: }1296=2\times2\times2\times2\times3\times3\times3\times3.$
$\displaystyle \therefore \sqrt{1296}=2\times2\times3\times3=36.$
$\displaystyle \text{Hence, }1296\text{ is a perfect square.}$
$\displaystyle \\$

$\displaystyle \textbf{To Find the Square Root of a Given Number by Division Method}$
$\displaystyle \text{1. Mark off the digits in pairs, starting from the unit digit. Each pair and}$
$\displaystyle \text{the remaining single digit, if any, is called a period.}$

$\displaystyle \text{2. Find the largest number whose square is equal to or less than the first}$
$\displaystyle \text{period. Take this number as both the divisor and the first digit of the quotient.}$

$\displaystyle \text{3. Subtract the square of this number from the first period and bring down}$
$\displaystyle \text{the next period to the right of the remainder to form the new dividend.}$

$\displaystyle \text{4. Double the quotient obtained so far to form the beginning of the new}$
$\displaystyle \text{divisor. Annex a suitable digit to it and take the same digit as the next}$
$\displaystyle \text{digit of the quotient. Choose the largest such digit so that the product}$
$\displaystyle \text{of the new divisor and this digit does not exceed the new dividend.}$

$\displaystyle \text{5. Subtract this product from the new dividend and bring down the next period.}$
$\displaystyle \text{Repeat the process until all the periods have been brought down.}$

$\displaystyle \text{The quotient so obtained is the required square root of the given number.}$
$\displaystyle \\$

Square root of numbers in decimal form

Method: Make the number of decimal places even, by affixing a zero, if necessary. Now, mark periods (starting from the right most digit) and find out the square root by the long-division method. Put the decimal point in the square root as soon as the integral part is exhausted.

Square root of numbers which are not perfect squares

$\displaystyle \textbf{Square Root of Fractions}$
$\displaystyle \text{For any positive real numbers }a\text{ and }b,\text{ we have:}$

$\displaystyle \text{(i) }\sqrt{ab}=\sqrt{a}\times\sqrt{b}.$
$\displaystyle \text{Example: }\sqrt{441}=21,\qquad\sqrt{1849}=43.$
$\displaystyle \therefore \frac{\sqrt{441}}{\sqrt{1849}}=\frac{21}{43}.$

$\displaystyle \text{(ii) }\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}},\qquad b>0.$
$\displaystyle \text{Example: }\sqrt{\frac{9}{16}} =\frac{\sqrt{9}}{\sqrt{16}}=\frac{3}{4}.$
$\displaystyle \\$

$\displaystyle \textbf{Cube of a Number}$
$\displaystyle \text{The cube of a number is that number raised to the power }3.$
$\displaystyle \text{Example: Cube of }2=2^3=2\times2\times2=8.$
$\displaystyle \\$

$\displaystyle \textbf{Perfect Cube}$
$\displaystyle \text{A natural number is called a perfect cube if it is the cube of some}$
$\displaystyle \text{natural number.}$
$\displaystyle \text{Example: }1^3=1,\qquad2^3=8,\qquad3^3=27,\text{ and so on.}$
$\displaystyle \therefore 1,\ 8\text{ and }27\text{ are perfect cubes.}$
$\displaystyle \\$

$\displaystyle \textbf{Cube Root}$
$\displaystyle \text{The cube root of a number }x\text{ is a number whose cube is }x.$
$\displaystyle \text{It is denoted by }\sqrt[3]{x}.$
$\displaystyle \text{Example: Since }5^3=5\times5\times5=125,\text{ therefore }\sqrt[3]{125}=5.$
$\displaystyle \\$

$\displaystyle \textbf{Method of Finding the Cube Root of Numbers}$
$\displaystyle \textbf{Cube Root of a Given Number by Prime Factorization Method}$
$\displaystyle \text{1. Resolve the given number into its prime factors.}$
$\displaystyle \text{2. Make groups of three equal prime factors.}$
$\displaystyle \text{3. Take one prime factor from each triplet and find their product.}$
$\displaystyle \text{The product so obtained is the cube root of the given number.}$

$\displaystyle \text{Example: Find the cube root of }17576.$
$\displaystyle 17576=2\times2\times2\times13\times13\times13$
$\displaystyle =2^3\times13^3.$
$\displaystyle \therefore \sqrt[3]{17576}=2\times13=26.$
$\displaystyle \\$

$\displaystyle \textbf{Cube Roots of Fractions and Decimals}$
$\displaystyle \text{For positive real numbers }a\text{ and }b,\text{ where }b\ne0,\text{ we have:}$
$\displaystyle \sqrt[3]{\frac{a}{b}} =\frac{\sqrt[3]{a}}{\sqrt[3]{b}}.$

$\displaystyle \text{Example 1: }\sqrt[3]{\frac{2197}{343}} =\frac{\sqrt[3]{2197}}{\sqrt[3]{343}}$
$\displaystyle =\frac{\sqrt[3]{13\times13\times13}} {\sqrt[3]{7\times7\times7}}=\frac{13}{7}.$

$\displaystyle \text{Example 2: }\sqrt[3]{19.683} =\sqrt[3]{\frac{19683}{1000}}$
$\displaystyle =\frac{\sqrt[3]{19683}}{\sqrt[3]{1000}} =\frac{27}{10}=2.7.$
$\displaystyle \\$

$\displaystyle \textbf{Square Roots by Using Tables}$
$\displaystyle \text{A table showing the square roots of all natural numbers from }1\text{ to }100$
$\displaystyle \text{is given below. Each square root is approximated to three decimal places.}$
$\displaystyle \text{Using this table, we can find the approximate square roots of numbers}$
$\displaystyle \text{and use them to find square roots of related larger numbers.}$

$\displaystyle \begin{array}{|c|c||c|c||c|c||c|c||c|c|} \hline x&\sqrt{x}&x&\sqrt{x}&x&\sqrt{x}&x&\sqrt{x}&x&\sqrt{x}\\ \hline 1&1.000&21&4.583&41&6.403&61&7.810&81&9.000\\ 2&1.414&22&4.690&42&6.481&62&7.874&82&9.055\\ 3&1.732&23&4.796&43&6.557&63&7.937&83&9.110\\ 4&2.000&24&4.899&44&6.633&64&8.000&84&9.165\\ 5&2.236&25&5.000&45&6.708&65&8.062&85&9.220\\ 6&2.449&26&5.099&46&6.782&66&8.124&86&9.274\\ 7&2.646&27&5.196&47&6.856&67&8.185&87&9.327\\ 8&2.828&28&5.292&48&6.928&68&8.246&88&9.381\\ 9&3.000&29&5.385&49&7.000&69&8.307&89&9.434\\ 10&3.162&30&5.477&50&7.071&70&8.367&90&9.487\\ 11&3.317&31&5.568&51&7.141&71&8.426&91&9.539\\ 12&3.464&32&5.657&52&7.211&72&8.485&92&9.592\\ 13&3.606&33&5.745&53&7.280&73&8.544&93&9.644\\ 14&3.742&34&5.831&54&7.348&74&8.602&94&9.695\\ 15&3.873&35&5.916&55&7.416&75&8.660&95&9.747\\ 16&4.000&36&6.000&56&7.483&76&8.718&96&9.798\\ 17&4.123&37&6.083&57&7.550&77&8.775&97&9.849\\ 18&4.243&38&6.164&58&7.616&78&8.832&98&9.899\\ 19&4.359&39&6.245&59&7.681&79&8.888&99&9.950\\ 20&4.472&40&6.325&60&7.746&80&8.944&100&10.000\\ \hline \end{array}$

$\displaystyle \textbf{Examples}$
$\displaystyle \sqrt{20}\approx4.472,\qquad\sqrt{54}\approx7.348,$
$\displaystyle \sqrt{83}\approx9.110,\qquad\sqrt{99}\approx9.950.$
$\displaystyle \\$

$\displaystyle \textbf{Cube Roots of Numbers Using Cube Root Tables}$
$\displaystyle \text{The table given below shows the approximate values of }\sqrt[3]{x},$
$\displaystyle \sqrt[3]{10x}\text{ and }\sqrt[3]{100x},\text{ where }x\text{ is a natural number}$
$\displaystyle \text{from }1\text{ to }100.\text{ The values are given correct to three decimal places.}$

$\displaystyle \begin{array}{|c|c|c|c||c|c|c|c||c|c|c|c|} \hline x&\sqrt[3]{x}&\sqrt[3]{10x}&\sqrt[3]{100x} &x&\sqrt[3]{x}&\sqrt[3]{10x}&\sqrt[3]{100x} &x&\sqrt[3]{x}&\sqrt[3]{10x}&\sqrt[3]{100x}\\ \hline 1&1.000&2.154&4.642&35&3.271&7.047&15.183&69&4.102&8.837&19.038\\ 2&1.260&2.714&5.848&36&3.302&7.114&15.326&70&4.121&8.879&19.129\\ 3&1.442&3.107&6.694&37&3.332&7.179&15.467&71&4.141&8.921&19.220\\ 4&1.587&3.420&7.368&38&3.362&7.243&15.605&72&4.160&8.963&19.310\\ 5&1.710&3.684&7.937&39&3.391&7.306&15.741&73&4.179&9.004&19.399\\ 6&1.817&3.915&8.434&40&3.420&7.368&15.874&74&4.198&9.045&19.487\\ 7&1.913&4.121&8.879&41&3.448&7.429&16.005&75&4.217&9.086&19.574\\ 8&2.000&4.309&9.283&42&3.476&7.489&16.134&76&4.236&9.126&19.661\\ 9&2.080&4.481&9.655&43&3.503&7.548&16.261&77&4.254&9.166&19.747\\ 10&2.154&4.642&10.000&44&3.530&7.606&16.386&78&4.273&9.205&19.832\\ 11&2.224&4.791&10.323&45&3.557&7.663&16.510&79&4.291&9.244&19.916\\ 12&2.289&4.932&10.627&46&3.583&7.719&16.631&80&4.309&9.283&20.000\\ 13&2.351&5.066&10.914&47&3.609&7.775&16.751&81&4.327&9.322&20.083\\ 14&2.410&5.192&11.187&48&3.634&7.830&16.869&82&4.344&9.360&20.165\\ 15&2.466&5.313&11.447&49&3.659&7.884&16.985&83&4.362&9.398&20.247\\ 16&2.520&5.429&11.696&50&3.684&7.937&17.100&84&4.380&9.435&20.328\\ 17&2.571&5.540&11.935&51&3.708&7.990&17.213&85&4.397&9.473&20.408\\ 18&2.621&5.646&12.164&52&3.733&8.041&17.325&86&4.414&9.510&20.488\\ 19&2.668&5.749&12.386&53&3.756&8.093&17.435&87&4.431&9.546&20.567\\ 20&2.714&5.848&12.599&54&3.780&8.143&17.544&88&4.448&9.583&20.646\\ 21&2.759&5.944&12.806&55&3.803&8.193&17.652&89&4.465&9.619&20.724\\ 22&2.802&6.037&13.006&56&3.826&8.243&17.758&90&4.481&9.655&20.801\\ 23&2.844&6.127&13.200&57&3.849&8.291&17.863&91&4.498&9.691&20.878\\ 24&2.884&6.214&13.389&58&3.871&8.340&17.967&92&4.514&9.726&20.954\\ \hline \end{array}$

$\displaystyle \textbf{Examples}$
$\displaystyle \sqrt[3]{5}\approx1.710,\qquad \sqrt[3]{50}\approx3.684,\qquad\sqrt[3]{500}\approx7.937.$
$\displaystyle \\$

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Class 8: Square Roots & Cube Roots (Lecture Notes)

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