Laws of Indices

'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: 27^4/3

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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