Answer
423.9k+ views
Hint: In this question, we will proceed by writing the given data and converting them to the desired way. Then substitute the terms in each other to form a relation between \[p,m,n\] to get the required answer. So, use this concept to reach the solution of the given problem.
Complete step-by-step answer:
Given that \[x\] varies as the \[{m^{th}}\] power of \[y\] i.e., \[y = {x^{\dfrac{1}{m}}} \Rightarrow x = {y^m}\]
And \[y\] varies as the \[{n^{th}}\] power of \[z\] i.e., \[z = {y^{\dfrac{1}{n}}} \Rightarrow y = {z^n}\]
\[
\Rightarrow y = {z^n} \\
\Rightarrow x = {\left( {{y^m}} \right)^n}{\text{ }}\left[ {\because x = {y^m}} \right] \\
\Rightarrow x = {z^{mn}}.................................\left( 1 \right) \\
\]
Also given that and \[x\] varies as the \[{p^{th}}\] power of \[z\]i.e., \[z = {x^{\dfrac{1}{p}}} \Rightarrow x = {z^p}.........................\left( 2 \right)\]
From equation (1) and (2), we have
\[ \Rightarrow {x^{mn}} = {x^p}\]
Since, the bases are equal we can equate the powers on both sides
\[
\Rightarrow mn = p \\
\therefore p = mn \\
\]
Thus, the correct option is C. \[p = mn\]
Note: Here, if \[a\] varies as the \[{b^{th}}\] power of \[c\], then it can be written as \[a = {c^{\dfrac{1}{b}}} \Rightarrow c = {a^b}\]. Whenever we have equal bases on both sides, we can equate the powers of the terms on both sides i.e., if \[{x^m} = {x^n}\] then \[m = n\].
Complete step-by-step answer:
Given that \[x\] varies as the \[{m^{th}}\] power of \[y\] i.e., \[y = {x^{\dfrac{1}{m}}} \Rightarrow x = {y^m}\]
And \[y\] varies as the \[{n^{th}}\] power of \[z\] i.e., \[z = {y^{\dfrac{1}{n}}} \Rightarrow y = {z^n}\]
\[
\Rightarrow y = {z^n} \\
\Rightarrow x = {\left( {{y^m}} \right)^n}{\text{ }}\left[ {\because x = {y^m}} \right] \\
\Rightarrow x = {z^{mn}}.................................\left( 1 \right) \\
\]
Also given that and \[x\] varies as the \[{p^{th}}\] power of \[z\]i.e., \[z = {x^{\dfrac{1}{p}}} \Rightarrow x = {z^p}.........................\left( 2 \right)\]
From equation (1) and (2), we have
\[ \Rightarrow {x^{mn}} = {x^p}\]
Since, the bases are equal we can equate the powers on both sides
\[
\Rightarrow mn = p \\
\therefore p = mn \\
\]
Thus, the correct option is C. \[p = mn\]
Note: Here, if \[a\] varies as the \[{b^{th}}\] power of \[c\], then it can be written as \[a = {c^{\dfrac{1}{b}}} \Rightarrow c = {a^b}\]. Whenever we have equal bases on both sides, we can equate the powers of the terms on both sides i.e., if \[{x^m} = {x^n}\] then \[m = n\].
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)