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Question: Evaluate the value of \(\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}}\) ....

Evaluate the value of xpxp+xq+1xp−q+1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} .

Explanation

Solution

Here, we are asked to find the value of xpxp+xq+1xp−q+1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} .
Using the property am−n=aman{a^{m - n}} = \dfrac{{{a^m}}}{{{a^n}}} , write xp−q{x^{p - q}} in fraction form in the second term of the given sum of two fractions.
Then, take LCM as per the requirements and solve the sum further to get the required answer.

Complete step-by-step answer:
Here, we are asked to find the value of xpxp+xq+1xp−q+1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} .
Now, in the denominator of the second term of the given sum, there is xp−q{x^{p - q}} .
Since, we know that, am−n{a^{m - n}} can be written as aman\dfrac{{{a^m}}}{{{a^n}}} , i.e. am−n=aman{a^{m - n}} = \dfrac{{{a^m}}}{{{a^n}}} .
So, using the above property of powers and exponents, we can write xp−q{x^{p - q}} as xpxq\dfrac{{{x^p}}}{{{x^q}}} .
Thus, xpxp+xq+1xp−q+1=xpxp+xq+1xpxq+1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} = \dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{\dfrac{{{x^p}}}{{{x^q}}} + 1}} .
Now, taking LCM in the denominator of the second term, we get
xpxp+xq+1xp−q+1=xpxp+xq+1xp+xqxq\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} = \dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{\dfrac{{{x^p} + {x^q}}}{{{x^q}}}}}
Also, 11a=a\dfrac{1}{{\dfrac{1}{a}}} = a
∴xpxp+xq+1xp−q+1=xpxp+xq+xqxp+xq\therefore \dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} = \dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{{{x^q}}}{{{x^p} + {x^q}}}
Again, taking LCM in the above sum will give
xpxp+xq+1xp−q+1=xp+xqxp+xq=1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} = \dfrac{{{x^p} + {x^q}}}{{{x^p} + {x^q}}} = 1
Thus, the value of xpxp+xq+1xp−q+1\dfrac{{{x^p}}}{{{x^p} + {x^q}}} + \dfrac{1}{{{x^{p - q}} + 1}} is 1.

Note: Some properties of powers and exponents are given as follows:
am×an=am+n an×bn=(a×b)n aman=am−n anbn=(ab)n 11a=a (am)n=am×n amn=amn an=a1n a−n=1an a0=1  {a^m} \times {a^n} = {a^{m + n}} \\\ {a^n} \times {b^n} = {\left( {a \times b} \right)^n} \\\ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} \\\ \dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n} \\\ \dfrac{1}{{\dfrac{1}{a}}} = a \\\ {\left( {{a^m}} \right)^n} = {a^{m \times n}} \\\ \sqrt[n]{{{a^m}}} = {a^{\dfrac{m}{n}}} \\\ \sqrt[n]{a} = {a^{\dfrac{1}{n}}} \\\ {a^{ - n}} = \dfrac{1}{{{a^n}}} \\\ {a^0} = 1 \\\
Remember these properties.