7th Class

     Positive Rational Number as Exponents     Let us consider 'a' be a positive rational number and \[x\] (which is \[\frac{m}{n}\]) be a positive rational exponent then it can also be written as \[{{a}^{x\left( i.e.{{a}^{\frac{m}{n}}} \right)}}\]or the  root of \[{{a}^{m}}.\] For example, \[{{4}^{\frac{3}{2}}}={{\left( {{4}^{3}} \right)}^{\frac{1}{2}}}={{\left( 4\times 4\times 4 \right)}^{\frac{1}{2}}}={{\left( 64 \right)}^{\frac{1}{2}}}=8\]    

     Law of Exponents     For any two rational numbers a and b and for any integer's m and n we have:  

• \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
• \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
• \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}={{\left( {{a}^{n}} \right)}^{m}}\]
• \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\]
• \[{{\left( \frac{a}{b} \right)}^{n}}=\frac{{{a}^{n}}}{{{b}^{n}}}\]
• If \[x\] is a rational number \[\left( x>0 \right)\] and a, b are rational exponents so that, a > b then \[{{x}^{a}}\div {{x}^{b}}={{x}^{a-b}}.\]
• If x is a rational number \[\left( x>0 \right)\] and a, b are rational exponents so that \[a<b\] then \[{{x}^{a}}\div {{x}^{b}}=\frac{1}{{{x}^{b-a}}}.\]
• If x is a rational number \[(x>0)\] and a, b and c are rational exponents then\[{{\left\{ {{\left( {{x}^{a}} \right)}^{b}} \right\}}^{c}}={{x}^{abc}}\]
• If \[x\] and y are rational numbers so that \[x>0,\text{ }y>0,\]and a is a rational exponent then \[{{x}^{a}}\times {{y}^{a}}={{\left( x\times y \right)}^{a}}.\]  

     Introduction   For any natural number p, the exponent is defined as: \[{{p}^{n}}=p\times p\times p.........\](up to n times), where \[{{p}^{n}}\] is called \[{{n}^{th}}\] power of p and read as "p raised to the power n".       Write 2401 in exponential form.   Solution: \[7\times 7\times 7\times 7={{7}^{4}}\]and \[\left( -7 \right)\times \left( -7 \right)\times \left( -7 \right)\times \left( -7 \right)=\left( -7 \right)4.\] Here \[{{7}^{4}}\]and \[{{\left( -7 \right)}^{4}}\]are said to be exponential form of 2401. Note: \[a{}^\circ =1\]for all rational number 'a', but in the case of negative exponent it is expressed as \[{{(a)}^{-n}}\]and is called the exponent of base "a" is \[-n\]and it can also be defined as \[{{a}^{-n}}=\frac{1}{{{a}^{n}}}.\]

     Introduction   We know that a mathematical statement of equality which involves one or more than one variables is called an equation. An equation in which variables are of one degree is called linear equation. If there Is only one variable, then the equation is said to be linear equation in one variable.          Linear Equation The general form of linear equation in one variable is \[ax+b=0,\] where a and b are constant. The general form of linear equation in two variable is \[ax+by+c=0,\] where a, b, c are constants, for example \[4x+5=5\] is a linear equation in one variable.           Solutions of the Linear Equations The real number which satisfies the given equation is called the solution of the equation, for example 6 is the root of the equation more...

     Application of Linear Equation   When you are solving the word problem you should follow the following steps: Step 1:   Read the problem carefully and specify the given and required parameters. Step 2:   Represent the unknown quantity by variables like x, y w....etc. Step 3:   Convert the mathematical statements into mathematical problem. Step 4:   Use the conditions to form an equation. Step 5:   Solve the equation for the unknown and check whether the solution satisfies the equating or not.     The sum of three consecutive multiples of 8 is 888. Which one of following options is the group of those numbers? (a) 504, 342 and 342                        (b) 234, 567 and 604 (c) 234, 564 and 905                         (d) 288, 296 and 304 (e) None of these     Answer: (d) Explanation   Let the more...

     Introduction In this chapter we will study about the comparison of two or more quantities. When we compare only two quantities of same kind, it is called ratio and more than two quantities is called proportion.          Ratio A ratio is a relation between two quantities of same kind. Comparison is made between the two quantities by considering what part of one quantity is that of the other quantity. The two quantities are called the terms of ratio. If \[x\]and y are two quantities of same kind then the ratio of \[x\] to \[y\]is \[x/y\]or \[x:y.\]It is represented by \[x:y.\]           Important Points Related to Ratio

• The first term of ratio is called antecedent and the second term is called the consequent.
• If \[\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=..............\]then each ratio is more...

     Proportion   It is the equality of two ratios i.e. if a: b = c: d, then ad = cd that implies product of extremes = product of means. Four quantities p, q, r, s are in proportion if ps = qr                   Important Points Related to Proportion If \[\frac{a}{b}=\frac{c}{d}\]then

• \[\frac{a+b}{b}=\frac{c+d}{d}\](componendo)
• \[\frac{a-b}{b}=\frac{c-d}{d}\](dividend)
• \[\frac{a+b}{a-b}=\frac{c+d}{c-d}\] (componendo and dividendo)
• If three numbers a, b, c are in continued proportion and written as a : b :: b: c then \[\frac{a}{b}=\frac{b}{c}\Rightarrow {{b}^{2}}=ac\Rightarrow b=\sqrt{ac}\] hence, b is called mean.  

          The Ratio between two quantities is 7: 9. If the first quantity is 511 then find the other quantity. (a) 655                                                  (b) 555 (c) 65                                                     (d) 656 (e) None of these     Answer: more...

       Rational Number                   Important Point Related to Rational Numbers

• Zero is a rational number because 0 can be written as \[\frac{0}{a}\] where a \[a\ne 0\]
• Every natural number is a rational number but the rational number may or may not be a natural number, for example \[\frac{2}{3}\] is a rational number which is not a natural number.
• Every whole number is a rational number but the rational number may or may not be a whole number, for example \[\frac{-2}{3}\] is a rational number which is not a whole number.
• Every integers is a rational number but the rational number may or may not be an integer, for example \[\frac{3}{5}\] is a rational number which is not an integer.  

          Types of Rational Number                 Rational numbers are more...

     Operation on Rational Numbers                   In this topic we will study about addition, subtraction, multiplication and division of rational numbers.                        Addition of Rational Numbers                 Step 1: Write the rational number in the standard form.                 Step 2: Make the denominator same by taking the LCM of denominators.                 Step 3: Write all the rational number with the same denominator.                 Step 4: Add the numerators                 Step 5: Write the numerator getting after addition on the denominator.                 Step 6: Reduce the rational number to lowest term.                                     Add \[2\frac{3}{5},\frac{-15}{13},\frac{-13}{-15},-4\frac{3}{5}\]                                 Solution:                                 Step 1: The standard form of given rational numbers are                 \[\frac{13}{5},\frac{-15}{13},\frac{13}{15},\frac{-23}{5}\]                                               Step 2: LCM of 5, 13, 15, 5 is 195                                 Step3: \[\frac{13}{5}=\frac{13\times more...

       Introduction                   We have already studied about the integers, the fractions, the decimals and their operations. We know that for a given integer a and b, their addition, subtraction and multiplication are always an integer but the result of division of an integer by a non-negative integer may or may not be an integer. In this chapter we will study about a new number system which is formed by the division of two integers.                 A number which is in the form of  \[\frac{a}{b}\] where and b are integers and \[b\ne 0\] is called a rational number, for example \[\frac{-1}{2},\frac{3}{-14},\frac{15}{7}\] are rational numbers.