Question

The value of \[\dfrac{{{x}^{a+b}}.{{x}^{b+c}}.{{x}^{c+a}}}{{{\left( {{x}^{a}}.{{x}^{b}}.{{x}^{c}} \right)}^{2}}}\] is: -
(a) \[{{x}^{2}}\]
(b) \[{{x}^{a+b+c}}\]
(c) \[{{x}^{abc}}\]
(d) \[{{x}^{0}}\]

Answer 1

Hint: Consider the numerator of the given expression and use the formula: - \[{{p}^{m}}\times {{p}^{n}}={{p}^{m+n}}\], to simplify the numerator. Now, consider the denominator and apply the formula: - \[{{\left( {{p}^{m}} \right)}^{n}}={{p}^{m\times n}}\] to simplify the denominator. Once both numerator and denominator are simplified, apply the formula, \[\dfrac{{{p}^{m}}}{{{p}^{n}}}={{p}^{m-n}}\] to get the answer.

Complete step-by-step solution
Here, we have been provided with the expression: - \[\dfrac{{{x}^{a+b}}.{{x}^{b+c}}.{{x}^{c+a}}}{{{\left( {{x}^{a}}.{{x}^{b}}.{{x}^{c}} \right)}^{2}}}\] and we have to find its value. Let us assume its value as ‘E’.
\[\Rightarrow E=\dfrac{{{x}^{a+b}}.{{x}^{b+c}}.{{x}^{c+a}}}{{{\left( {{x}^{a}}.{{x}^{b}}.{{x}^{c}} \right)}^{2}}}\] - (1)
Now, considering the numerator of expression (1), we have,
\[\Rightarrow \] Numerator = \[{{x}^{a+b}}.{{x}^{b+c}}.{{x}^{c+a}}\]
We know that, \[{{p}^{m}}\times {{p}^{n}}={{p}^{m+n}}\], therefore we have,
\[\Rightarrow \] Numerator = \[{{x}^{a+b+b+c+c+a}}\]
\[\Rightarrow \] Numerator = \[{{x}^{2a+2b+2c}}\]
Now, considering the denominator of expression (1), we have,
\[\Rightarrow \] Denominator = \[{{\left( {{x}^{a+b+c}} \right)}^{2}}\]
Now, applying the formula, \[{{\left( {{p}^{m}} \right)}^{n}}={{p}^{m\times n}}\], we get,
\[\Rightarrow \] Denominator = \[{{x}^{\left( a+b+c \right)\times 2}}\]
\[\Rightarrow \] Denominator = \[{{x}^{2\left( a+b+c \right)}}\]
Therefore, substituting the obtained values of numerator and denominator in expression (1), we have,
\[\Rightarrow E=\dfrac{{{x}^{2\left( a+b+c \right)}}}{{{x}^{2\left( a+b+c \right)}}}\]
Applying the formula, \[\dfrac{{{p}^{m}}}{{{p}^{n}}}={{p}^{m-n}}\], we get,
\[\Rightarrow E={{x}^{2\left( a+b+c \right)-2\left( a+b+c \right)}}\]
Cancelling the like terms, we get,
\[\Rightarrow E={{x}^{0}}\]
Hence, option (d) is the correct answer.

Note: One may note that we can solve the question by only simplifying either numerator or denominator but here we have simplified both of them so that we can see the use of all the important formulas of the topic ‘exponents and powers’. We must remember some basic formulas of the topic ‘exponents and powers’ like: - \[{{p}^{m}}\times {{p}^{n}}={{p}^{m+n}}\], \[{{p}^{m}}\div {{p}^{n}}={{p}^{m-n}}\] and \[{{\left( {{p}^{m}} \right)}^{n}}={{p}^{m\times n}}\]. These formulas are used everywhere.