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.

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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