# 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$(\sqrt{x})^{2}=x$so 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. The$x^{\frac{1}{y}}$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 $2^{\frac{1}{4}}$ < $3^{\frac{1}{5}}$