Surds and Indices Shortcuts, Tricks, PDF and Formulas

Surds and Indices Points to Remember - Page 2
Surds and Indices Examples - Page 3
Surds and Indices Important Questions - Page 5

Important Formulas - Surds and Indices

  • An integer is a whole number (positive, negative or zero). A rational number is one that can be expressed as a fraction \frac{a}{b}, where a and b are integers. All integers, fractions and terminating or recurring decimals are rational.
  • An irrational number cannot be expressed in the form \frac{a}{b}, where a and b are integers. Examples of irrational numbers are \pi,\sqrt{2},\sqrt{3}\;\;and\:4\sqrt{5}\:(4\sqrt{5}\:means\:4\times \sqrt{5}).
  • Real numbers are numbers that can be represented by points on the number line. Real numbers include both rational and irrational number.
  • A surd is an irrational number involving a root. The numbers  \sqrt{3},4\sqrt{5}\:and\:\sqrt[3]{7} are examples of surds. Numbers such as  \sqrt{16}and \sqrt[3]{8} are not surds because they are equal to rational numbers.  \sqrt{16} =4 and root of ‘3’ or ‘root 3’. Note that we cannot take the square root of a negative number.
  • Like surds involve the square root of the same number. Only like surds can be added or subtracted. For example,  3\sqrt{2}+4\sqrt{2}=7\sqrt{2}\:but\: 3\sqrt{2}+4\sqrt{3}=3\sqrt{2}+4\sqrt{3}.
  • Surds of the form \sqrt{x} can be simplified if the number beneath the square root sign has a factor that is a perfect square. For example, \sqrt{8}=\sqrt{4\times 2}=\sqrt{4}\times \sqrt{2}=2\sqrt{2}.
  • The following rules can be used when multiplying or dividing surds.
    (\sqrt{x})^{2}=\sqrt{x^{2}}=x
    \sqrt{x}\times \sqrt{y}=\sqrt{xy}
    \frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}
  • Rationalising the denominator of a surd means changing the denominator so that is a rational number. To rationalize the denominator of a surd such as\frac{\sqrt{2}}{\sqrt{3}} we use the result that(\sqrt{x})^{2}=xso if we multiply the denominator by\sqrt{3}it will be rational. if we multiply the demoniator by\sqrt{3}we must also multiply the numerators by\sqrt{3}so that\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{2}}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{6}}{\sqrt{3}}
  • Fractional indices may be used to express roots.x^{\frac{1}{n}}=\sqrt[n]{x}\:\:and\:\:x^{\frac{m}{n}}=(\sqrt[n]{x})^{m}
  • Make sure you can use your calculator to find powers and roots. Thex^{\frac{1}{y}}or \sqrt[x]{\:}button allows you to find roots.
    For example:  \sqrt[5]{7776}=(7776)^{\frac{1}{5}}

INDICES

  • When the powers are fraction , then we can compare the indices in following manner.
    for example :find which is greather 2^{\frac{1}{4}} or 3^{\frac{1}{5}}
    step-1 : find the L.C.M of the denominator of the fraction i.e 4,5 = 20
    step-2 : find powers with the L.C.M 2^{\frac{1}{4}\times 20} OR 3^{\frac{1}{5}\times 20} = 2^{5} AND 3^{4}
    step-3 : compare the result obtained in setp-2 : i.e as 32 < 81 we can say 2^{\frac{1}{4}} < 3^{\frac{1}{5}}

 

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