POWER INDICES AND SURDS RULES OR FORMULAS

Importance : 1 or 2 questions from ‘Surds and Indices’ have essentially been asked in every exam. In order to accuracy in your calculations, you will require complete practice of this chapter.

Scope of questions : Asked questions are based on basic concepts, completely arithmetic and without language like to evaluate /simply, greatest/lowest number, increasing/ decreasing order, square, cube, square root, cube root and higher powers starting from easier to tougher levels.

Way to success : Note that practice to solve these questions with full concentration and accuracy is essential. Only because of small mistake or not understanding , the basic concepts many students are unable to solve these questions.

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INDICES
In seventeenth century, a French mathematician Reni Dakata’s multiplied a number several times and showed the obtained product by a special rule, which is called ‘indices’ and the converse of indices is called surds.

Rule 1 : If any number is multiplied by the same number ‘n’ times, then,

a × a × a × a ………. × a (n times) = aⁿ

(i) where n and a are real numbers.(including fractions)

(ii) a is called base.

(iii) n is called indices.

Rule 2 : a^m × aⁿ = a^m+n

and a^m × aⁿ × a^p = a^m+n+p

While multiplying, If base is same then powers get added.

Rule 3 : While multiplying , if bases are different but powers are same then,

a^x × b^x × c^x = (abc)^x

Rule 4 : While dividing, if base is same, then powers get subtracted , as

a^m ÷ aⁿ = a^{m – n}

Rules 5 : If there is negative indices on a number, then

$${ a }^{ -m }=\frac { 1 }{ { a }^{ m } } \quad or,\quad { a }^{ m }=\frac { 1 }{ { a }^{ -m } } $$

Rule 6 : If there are indices on indices, then indices are multiplied. As,

(i) $${ \left( { a }^{ m } \right) }^{ n }={ a }^{ mn }$$

(ii) $${ \left( { a }^{ m } \right) }^{ \frac { 1 }{ n } }={ a }^{ \frac { m }{ n } }$$

(iii) $${ \left\{ { \left( { a }^{ m } \right) }^{ n } \right\} }^{ p }={ a }^{ mnp }$$

Rule 7 : (i) $${ { a }^{ m } }^{ n }\neq { \left( { a }^{ m } \right) }^{ n }$$

(ii) $${ { a }^{ m } }^{ \frac { 1 }{ n } }\neq { \left( { a }^{ m } \right) }^{ \frac { 1 }{ n } }$$

(iii) $${ { a }^{ m } }^{ { n }^{ p } }\neq { \left\{ { \left( { a }^{ m } \right) }^{ n } \right\} }^{ p }$$

Rule 8 : Indices as fraction.

(i) $${ \left( \frac { a }{ b } \right) }^{ n }=\quad \frac { { a }^{ m } }{ { b }^{ m } } $$

(ii) $${ \left( \frac { a }{ b } \right) }^{ -m }=\quad { \left( \frac { b }{ a } \right) }^{ m }$$

Rule 9 : If a^x = a^y , then x = y and if xⁿ = yⁿ , then x = y

Rule 10 : If the indices on any number is zero, the value of that number is 1, as

x⁰ = 1, 5⁰ = 1, 10⁰ = 1, (5000)⁰ = 1

Rule 11 : If ‘a’ is a rational number and n is a positive integer, then $$nth\quad root\quad of\quad ‘a’,{ a }^{ \frac { 1 }{ n } }or\quad \sqrt [ n ]{ a } $$ is an irrational number,

$$\sqrt [ n ]{ a } \quad is\quad called\quad surd\quad of\quad n\quad indices,\quad $$

$$it\quad means\quad \sqrt [ n ]{ a } \quad is\quad a\quad surd$$

Where (i) ‘a’ is a rational number.

(ii) ‘n’ is a positive integer.

$$(iii)\quad \sqrt [ n ]{ a } \quad is\quad an\quad irrational\quad number.$$

Rule 12 : $$If\quad \sqrt [ n ]{ a } is\quad a\quad surd,$$ then n is called surd indices and a is called ‘Radicand’ . Every surd can be an irrational number, but every irrational number can not be a surd.

Rule 13 : Mixed Surds – A surd having a rational co-efficient other than unity is called a mixed surd.

Rule 14 : Pure Surd – The surds whose one factor is 1 and other factor is an irrational number, then that type of surd or the surd which is completely under radical sign.

Rule 15 : Similar Surds – The surds whose irrational factor is same, that is called similar surds.

Rule 16 : Irrational numbers as −  √2 ,  √3 ,  √5 ,  √7 …….. etc. have infinite recurring decimals.

Rule 17 : $$\sqrt [ n ]{ a } ={ \left( a \right) }^{ \frac { 1 }{ n } }$$

Rule 18 : $${ \left( \sqrt [ n ]{ a } \right) }^{ n }=a$$

Rule 19 : $$\sqrt [ n ]{ ab } =\sqrt [ n ]{ a } \times \sqrt [ n ]{ b } ={ \left( a \right) }^{ \frac { 1 }{ n } }\times { (b) }^{ \frac { 1 }{ n } }$$

Rule 20 : $$\sqrt [ n ]{ \sqrt [ n ]{ a } } ={ \left( { \left( a \right) }^{ \frac { 1 }{ n } } \right) }^{ \frac { 1 }{ n } }={ a }^{ { n }^{ \frac { 1 }{ 2 } } }$$

Rule 21 : $$\sqrt [ n ]{ \frac { a }{ b } } =\frac { \sqrt [ n ]{ a } }{ \sqrt [ n ]{ b } } ={ \left( \frac { a }{ b } \right) }^{ \frac { 1 }{ n } }$$

Rule 22 : $$\sqrt [ m ]{ \sqrt [ n ]{ a } } =\sqrt [ mn ]{ a }  $$

Rule 23 : $$\sqrt { x\sqrt { x\sqrt { x\sqrt { x…….n\quad times } } } } ={ x }^{ \left( 1-\frac { 1 }{ { x }^{ n } } \right) }$$

Rule 24 : $$If\quad \sqrt { x-\sqrt { x-\sqrt { x-\quad …\infty } } } ,$$ where x n(n + 1)

then, $$\sqrt { x-\sqrt { x-\sqrt { x-\quad …\infty } } } =n$$

Rule 25 : $$If\quad \sqrt { x+\sqrt { x+\sqrt { x+\quad …\infty } } } $$ where x =n(n+1)

then, $$\sqrt { x+\sqrt { x+\sqrt { x+\quad …\infty } } } =(n+1)$$

Rule 26 : $$\sqrt [ a ]{ b } ,\sqrt [ x ]{ y } ,\sqrt [ n ]{ m } ,\sqrt [ p ]{ q } $$

To find smallest or greatest out of these, we should equate all the indices and compare the base.

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