Laws of Indices

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'Indices' is the plural form of 'index'. And, index means the power of any number or variable term, which is called a base. Power or index is also known as an exponent.



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Coefficient, base and index

 

There are certain rules which are used to solve the problems of indices. These rules are called the laws of indices. Some laws of indices are as follows: 



Laws of Indices


Law’s of indices: 1) xa × xb = xa + b 2) xa ÷ xb = xa – b 3) (xa)b = xa × b 4) (xy)a = xa × ya 5) "x" ^"-a" = "1" /〖" x" 〗^"a" 6) "1" /"x" ^"-a" = "x" ^"a" 7) ("x" /"y" )^"-a" = ("y" /"x" )^"a" 8) If xa = ya then x = y 9) If xa = xb then a = b 10) x0 = 1 11) √("a" ) = a1/2 12) √("3" &"a" ) = a1/3 13) √("n" &"a" ) = a1/n 14) If xn = m then x = √("n" &"m" )

Problems involving laws of indices


Example 1: Find the value of: 274/3

 

Solution:

 

Here, 

〖"27" 〗^("4" /"3" ) = ("3" ^"3" )^("4" /"3" ) = "3" ^("3×" "4" /"3" ) = "3" ^"4" = "81"


Example 2: Find the value of: (2401)-1/4

 

Solution:

 

Here,

〖"(2401)" 〗^("- " "1" /"4" ) = 〖"(" "7" ^"4" ")" 〗^("- " "1" /"4" ) = "7" ^("4 × - " "1" /"4" ) = 7-1 = "1" /"7"

Example 3: Find the value of: 

[√("3" &"64" ) " ÷ " ("1" /"125" )^("-1" /"3" ) ]^"2"

 

Solution:

 

Here,

[√("3" &"64" ) " ÷ " ("1" /"125" )^("-1" /"3" ) ]^"2" = ["4" ^("3 ×" "1" /"3" ) " ÷ " ("125" )^("1" /"3" ) ]^"2" = ["4 ÷ " "5" ^("3 × " "1" /"3" ) ]^"2" = ["4 ÷ 5" ]^"2" = ["4" /"5" ]^"2" = "16" /"25"

       

Example 4: Find the value of: 

("81" /"16" )^("-3" /"4" ) " × " [("25" /"9" )^("-3" /"2" ) " ÷ " ("5" /"2" )^"-3" ]


Solution:

 

Here,

("81" /"16" )^("-3" /"4" ) " × " [("25" /"9" )^("-3" /"2" ) " ÷ " ("5" /"2" )^"-3" ] = ("16" /"81" )^("3" /"4" ) " × " [("9" /"25" )^("3" /"2" ) " ÷ " ("2" /"5" )^"3" ] = ("2" /"3" )^("4 × " "3" /"4" ) " × " [("3" /"5" )^("2 × " "3" /"2" ) " ÷ " "8" /"125" ] = ("2" /"3" )^"3" " × " [("3" /"5" )^"3" " × " "125" /"8" ] = "8" /"27" " ×" "27" /"125" " × " "125" /"8" = 1

 Example 5: Find the value of: 

("x" ^"a" /"x" ^"-b" )^"a-b" " ." ("x" ^"b" /"x" ^"-c" )^"b-c" "." ("x" ^"c" /"x" ^"-a" )^"c-a"


Solution:

 

Here, 

= ("x" ^"a" /"x" ^"-b" )^"a-b" × ("x" ^"b" /"x" ^"-c" )^"b-c" × ("x" ^"c" /"x" ^"-a" )^"c-a" = "x" ^("a+b" )"(a-b)" × "x" ^("b+c" )"(b-c)" × "x" ^("c+a" )"(c-a)" = "x" ^("a" ^"2" "-" 〖" b" 〗^"2" ) × "x" ^("b" ^"2" "- " "c" ^"2" ) × "x" ^("c" ^"2" "-" 〖" a" 〗^"2" ) = "x" ^("a" ^"2" "- " "b" ^"2" "+ " "b" ^"2" "- " "c" ^"2" "+ " "c" ^"2" "- " "a" ^"2" ) = "x" ^"0" = 1

 


Example 6: Simplify: 

("x" ^"a" "÷" "x" ^"b" )^("a" ^"2" "+ab+" "b" ^"2" ).("x" ^"b" "÷" "x" ^"c" )^("b" ^"2" "+bc+" "c" ^"2" ).("x" ^"c" "÷" "x" ^"a" )^("c" ^"2" "+ca+" "a" ^"2" )


Solution:

 

Here,         

= ("x" ^"a" "÷" "x" ^"b" )^("a" ^"2" "+ab+" "b" ^"2" ).("x" ^"b" "÷" "x" ^"c" )^("b" ^"2" "+bc+" "c" ^"2" ).("x" ^"c" "÷" "x" ^"a" )^("c" ^"2" "+ca+" "a" ^"2" ) = "x" ^(("a-b" )"(" "a" ^"2" "+ab+" "b" ^"2" ")" )."x" ^(("b-c" )"(" "b" ^"2" "+bc+" "c" ^"2" ")" ) "." "x" ^(("c-a" )"(" "c" ^"2" "+ca+" "a" ^"2" ")" ) = "x" ^("a" ^"3" "-" "b" ^"3" ) " × " "x" ^("b" ^"3" "-" "c" ^"3" ) " × " "x" ^("c" ^"3" "-" "a" ^"3" ) = "x" ^("a" ^"3" "-" "b" ^"3" "+" "b" ^"3" "-" "c" ^"3" "+" "c" ^"3" "-" "a" ^"3" ) = "x" ^"0" = 1 

Example 7: Simplify: 

√("ab" &"x" ^"a" /"x" ^"b" ) " × " √("bc" &"x" ^"b" /"x" ^"c" ) " × " √("ca" &"x" ^"c" /"x" ^"a" )


Solution:

 

Here, 

√("ab" &"x" ^"a" /"x" ^"b" ) " × " √("bc" &"x" ^"b" /"x" ^"c" ) " × " √("ca" &"x" ^"c" /"x" ^"a" ) = ("x" ^"a" /"x" ^"b" )^("1" /"ab" ) " × " ("x" ^"b" /"x" ^"c" )^("1" /"bc" ) " × " ("x" ^"c" /"x" ^"a" )^("1" /"ca" ) = "x" ^("1" /"b" )/"x" ^("1" /"a" ) " × " "x" ^("1" /"c" )/"x" ^("1" /"b" ) " × " "x" ^("1" /"a" )/"x" ^("1" /"c" ) = "x" ^("1" /"b" " + " "1" /"c" " + " "1" /"a" " - " "1" /"a" " - " "1" /"b" " - " "1" /"c" ) = "x" ^"0" = 1



Example 8: Simplify: 

"1" /("1+ " "x" ^"a-b" "+ " "x" ^"a-c" ) "+ " "1" /("1+ " "x" ^"b-c" "+" "x" ^"b-a" ) "+ " "1" /("1+ " "x" ^"c-a" "+ " "x" ^"c-b" )


Solution:

 

Here, 

     "1" /("1+ " "x" ^"a-b" "+ " "x" ^"a-c" ) "+ " "1" /("1+ " "x" ^"b-c" "+" "x" ^"b-a" ) "+ " "1" /("1+ " "x" ^"c-a" "+ " "x" ^"c-b" ) = "x" ^"-a" /("(1+ " "x" ^"a-b" "+ " "x" ^"a-c" )"x" ^"-a" ) " + " "x" ^"-b" /("(1+ " "x" ^"b-c" "+" "x" ^"b-a" )"x" ^"-b" ) " + " "x" ^"-c" /("(1+ " "x" ^"c-a" "+ " "x" ^"c-b" )"x" ^"-c" ) = "x" ^"-a" /("x" ^"-a" " + " "x" ^"-b" "+ " "x" ^"-c" ) "+ " "x" ^"-b" /("x" ^"-b" " + " "x" ^"-c" "+" "x" ^"-a" ) "+ " "x" ^"-c" /("x" ^"-c" " + " "x" ^"-a" "+ " "x" ^"-b" ) = ("x" ^"-a" "+ " "x" ^"-b" "+ " "x" ^"-c" )/("x" ^"-a" "+ " "x" ^"-b" "+ " "x" ^"-c" ) = 1

Example 9: Simplify: 

("x" ^"a+b" /"x" ^"c" )^"a-b" " × " ("x" ^"b+c" /"x" ^"a" )^"b-c" " × " ("x" ^"c+a" /"x" ^"b" )^"c-a"


Solution:

 

Here,

("x" ^"a+b" /"x" ^"c" )^"a-b" " × " ("x" ^"b+c" /"x" ^"a" )^"b-c" " × " ("x" ^"c+a" /"x" ^"b" )^"c-a" = "x" ^("a+b" )"(a-b)" /"x" ^"c(a-b)" " × " "x" ^("b+c" )"(b-c)" /"x" ^"a(b-c)" " × " "x" ^("c+a" )"(c-a)" /"x" ^"b(c-a)" = "x" ^("a" ^"2" "-" "b" ^"2" )/"x" ^"ac-bc" " × " "x" ^("b" ^"2" "-" "c" ^"2" )/"x" ^"ab-ac" " × " "x" ^("c" ^"2" "-" "a" ^"2" )/"x" ^"bc-ab" = "x" ^("a" ^"2" "-" 〖" b" 〗^"2" "+" 〖" b" 〗^"2" "-" 〖" c" 〗^"2" "+" 〖" c" 〗^"2" "-" 〖" a" 〗^"2" )/"x" ^"ac – bc + ab – ac + bc - ab" = "x" ^"0" /"x" ^"0" = "1"
     

 

Example 10: Find the value of: 

("2" ^"n+1" " × " "2" ^("n-1" )("n+1" ) )/("2" ^"n(n-1)" " × " "4" ^"n+1" )


Solution:

 

Here,

("2" ^"n+1" " × " "2" ^("n-1" )"(n+1)" )/("2" ^"n(n-1)" " × " "4" ^"n+1" ) = ("2" ^"n+1" " × " "2" ^("n" ^"2" "-1" ))/("2" ^("n" ^"2" "-n" ) " × " "2" ^"2n+2" ) = ("2" ^("n+1+" "n" ^"2" "-1" ) " " )/("2" ^("n" ^"2" "-n+2n+2" ) " " ) = "2" ^("n+" "n" ^"2" "-" "n" ^"2" "-n-2" ) = "2" ^"-2" = "1" /"2" ^"2" = "1" /"4"

  

 

If you have any question or problems regarding the law’s  of indices, you can ask here, in the comment section below.


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4 comments:

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