Question

How do you simplify ${\left( {xy} \right)^{ - 1}}$ ?

Answer 1

Hint:The two mathematical terms used to simplify algebraic problems are exponents and powers. Power is used to represent how many times a number should be utilized in a given multiplication. Whereas, exponents are either positive or negative numbers which represent the power to which the base number is raised. Here in this question, we are supposed to simplify ${\left( {xy} \right)^{ - 1}}$ which can easily be done by using a few rules and laws of exponents.

Complete step by step answer:
Before starting to simplify the given problem, we should first know two terms: power and exponents.
Power: It is an expression which represents repeated multiplication of the same number.
Exponent: It is a quantity which represents the power to which the number is raised.
There are few laws of exponent, which are as follows:
Multiplication Law: ${a^n} \times {a^m} = {a^{n + m}}$
Division Law: $\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}$
Negative Exponent Law: ${a^{ - m}} = \dfrac{1}{{{a^m}}}$
Some rules of exponent are:
${a^0} = 1$
$\Rightarrow{\left( {{a^m}} \right)^n} = {a^{nm}}$
$\Rightarrow{a^m} \times {b^m} = {\left( {ab} \right)^m}$
$\Rightarrow\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}$
$\Rightarrow{a^1} = a$
Given expression is ${\left( {xy} \right)^{ - 1}}$.In order to simplify the given expression, we will have to use laws and rules of exponents. As through the negative exponent rule we know that, ${a^{ - m}} = \dfrac{1}{{{a^m}}}$. Using this law of exponents, we can say that,
${\left( {xy} \right)^{ - 1}} = \dfrac{1}{{{{\left( {xy} \right)}^1}}}$-----(1)
By using ${a^1} = a$ rule of exponent in equation (1) we can conclude that,
$\dfrac{1}{{{{\left( {xy} \right)}^1}}} = \dfrac{1}{{xy}}$

Hence, the required answer is $\dfrac{1}{{xy}}$.

Note:While solving such types of problems, students should carefully perform calculations. They should keep in mind the rules and the laws of exponents as they help a lot to easily simplify the problem. Some of the common mistakes which students make while solving exponents are in product rule and in quotient rule.