Question

How do you simplify \[{\left( {3{x^4}} \right)^{ - 2}}\] ?

Answer 1

Hint:In this question, we will solve by breaking the brackets and then we will get the expression in the exponents for number and each variable and applying the formulas \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\] , and then we will apply the exponent identity \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\] , and then further simplify the expression using the identity \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] to get the required result.

Complete step by step answer:Exponents are defined as when an expression or a statement of specific natural numbers are represented as a repeated power by multiplication of its units then the resulting number is called as an exponent. The resulting set of numbers are the same as the original sequence.
Given expression \[{\left( {3{x^4}} \right)^{ - 2}}\] ,
Now the given expression can be written by breaking the brackets as,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = {\left( {3 \times {x^4}} \right)^{ - 2}}\] ,
Now rewrite the expression using identity \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\] , we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = {\left( 3 \right)^{ - 2}} \times {\left( {{x^4}} \right)^{ - 2}}\] ,
Now using the exponent identity, \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\] for each term we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \left( {{3^{ - 2}}} \right) \times \left( {{x^{4 \times - 2}}} \right)\] ,
Now simplifying in the powers we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \left( {{3^{ - 2}}} \right) \times \left( {{x^{ - 8}}} \right)\] ,
Now using the identity, \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] , we get,
Here for the first term i.e.,${3^{ - 2}}$ \[a = 3\] and \[m = 2\] , and for the variable term i.e.,${x^{ - 8}}$$a = x$and$m = 8$, now substituting these values in the identity we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \left( {\dfrac{1}{{{3^2}}}} \right) \times \left( {\dfrac{1}{{{x^8}}}} \right)\] ,
Now simplifying by multiplying the terms we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \dfrac{1}{{{3^2}{x^8}}}\] ,
Now simplifying the expression we get,
 \[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \dfrac{1}{{9{x^8}}}\] ,
So the given expression is equal to \[\dfrac{1}{{9{x^8}}}\] .
Final Answer:
 \[\therefore \] The simplified form of given expression \[{\left( {3{x^4}} \right)^{ - 2}}\] will be equal to \[\dfrac{1}{{9{x^8}}}\] .

Note:
There are various laws of exponents we should remember and practise in order to solve and understand the exponential concept. The following are some of the exponent laws:
 \[{a^0} = 1\] ,
 \[{a^m} \times {a^n} = {a^{m + n}}\] ,
 \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] ,
 \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] ,
 \[{a^m} \times {b^m} = {\left( {ab} \right)^m}\] ,
 \[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\] .