Question

By what number should \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] be divided to get \[{\left( {\dfrac{2}{3}} \right)^2}\] ?

Answer 1

Hint: In the above question, we are given a number as \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] . We have to find an another number such that dividing \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] by that number gives us the result equal to the number \[{\left( {\dfrac{2}{3}} \right)^2}\] . In order to approach the solution, let us assume here that the number which will be the divisor be \[x\] .Now we can use the properties of exponents and powers and the Euclid’s division algorithm to find the value of \[x\] . To find the value of \[x\] , we can divide \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] by the given quotient, that is \[{\left( {\dfrac{2}{3}} \right)^2}\].

Complete step by step answer:
Given number is \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\]. We have to find an another number such that dividing \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] by that number gives us the number \[{\left( {\dfrac{2}{3}} \right)^2}\]. Let the other number, i.e. the divisor be \[x\].Now we have to find \[x\] such that,
\[ \Rightarrow {\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}} \div x = {\left( {\dfrac{2}{3}} \right)^2}\]

We can write the above equation as,
\[ \Rightarrow {\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}} \cdot \dfrac{1}{x} = {\left( {\dfrac{2}{3}} \right)^2}\]
Multiplying both sides by \[x\] ,
\[ \Rightarrow {\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}} = {\left( {\dfrac{2}{3}} \right)^2} \cdot x\]
Dividing both sides by \[{\left( {\dfrac{2}{3}} \right)^2}\] , we get
\[ \Rightarrow {\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}} \div {\left( {\dfrac{2}{3}} \right)^2} = x\]
We can write the above equation as,
\[ \Rightarrow {\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}} \cdot {\left( {\dfrac{3}{2}} \right)^2} = x\]
Since, \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]

Therefore,
\[ \Rightarrow {\left( {\dfrac{2}{{ - 3}}} \right)^3} \cdot {\left( {\dfrac{3}{2}} \right)^2} = x\]
Expanding the equation, we can write
\[ \Rightarrow \dfrac{{2 \times 2 \times 2}}{{\left( { - 3} \right) \times \left( { - 3} \right) \times \left( { - 3} \right)}} \times \dfrac{{3 \times 3}}{{2 \times 2}} = x\]
Cancelling the equal terms, we get
\[ \Rightarrow \dfrac{2}{{\left( { - 3} \right)}} = x\]
\[ \therefore x = - \dfrac{2}{3}\]

Therefore, the number \[{\left( {\dfrac{3}{{ - 2}}} \right)^{ - 3}}\] should be divided by the divisor \[ - \dfrac{2}{3}\] to get the quotient \[{\left( {\dfrac{2}{3}} \right)^2}\].

Note: Exponentiation is a mathematical operation, written as \[{a^n}\] , involving two numbers, the base \[a\] and the exponent or power \[n\] . When \[n\] is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, \[{a^n}\] is the product of multiplying the base \[a\]by \[n\] times. That is,
\[{a^n} = n \times n \times n \times ... \times n\] ( \[n\] times)