   Question Answers

# Find the natural number n, such that ${2^8} + {2^{10}} + {2^n}$ is a perfect square.  Hint: In this question first try to convert ${2^8} + {2^{10}} + {2^n}$ in ${\left( {a + b} \right)^2}$. As we know that ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ as in the equation their is also three term hence by comparing and some analysis we find that $a = {2^4}, b = {2^5}$ or ${({2^4} + {2^5})^2}$ = ${2^8} + {2^{10}} + {2^n}$ by comparing we will find n.

As in the given question ${2^8} + {2^{10}} + {2^n}$ we have to find the value of n for which it is a perfect square.
For this we have to try to make it like ${a^2} + {b^2} + 2ab$
Hence, ${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2^n}$
As for we have change it as ${2^n} = 2ab$
So that is equal to
${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2.2^{n - 1}}$
So form this we know that as $a = {2^4},b = {2^5}$ ,
hence
${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
${({2^4} + {2^5})^2}$ = ${2^8} + {2^{10}} + {2.2^{4 + 5}}$
Hence by comparing with real equation
${2^8} + {2^{10}} + {2.2^{4 + 5}}$ = ${\left( {{2^4}} \right)^2} + {\left( {{2^5}} \right)^2} + {2^n}$
From this ${2^8},{2^{10}}$ is cancel out from both side the remaining term is
${2^{10}} = {2^n}$
By comparing the power we get $n = 10$
Hence $n = 10$ will be the answer for making this as perfect square

Note: Sometimes the question will frame like that ${3^8} + {3^{10}} + {2.3^n}$ this question is same as the above question solve it , as in the above the question will solve.
Whenever we have to give question like this just try to make it as ${\left( {a + b} \right)^2}$ as in the question that the power of $2$ is $8, 10$ it is multiply of $2$ or written as ${({2^4})^2}, {({2^5})^2}$ and n is unknown.
View Notes
Some Natural Phenomena  Overview of Natural Resource Management  Table of 28 - Multiplication Table of 28  The Difference Between an Animal that is A Regulator and One that is A Conformer  CBSE Class 7 Maths Chapter 9 - Rational Numbers Formulas  How to Find Square Root of a Number  Table of 7 - Multiplication Table of 7  CBSE Class 9 Maths Chapter 1 - Number Systems Formulas  CBSE Class 7 Maths Formulas  CBSE Class 7 Maths Chapter 1 - Integers Formulas  CBSE Class 8 Science Some Natural Phenomena Worksheets  Important Questions for CBSE Class 7 English Honeycomb Chapter 4 - The Ashes That Made Trees Bloom  Important Questions for CBSE Class 7 Social Science Our Environment Chapter 6 - Natural Vegitation and Wildlife  Important Questions for CBSE Class 9 Science Chapter 14 - Natural Resources  Important Questions for CBSE Class 10 Science Chapter 16 - Management of Natural Resources  Important Questions for CBSE Class 6 Hindi Durva Chapter 28 - Gadha aur Siyar  Important Questions for CBSE Class 7 Maths  Important Questions for CBSE Class 7 Science Chapter 7 - Weather, Climate and Adaptations of Animals to Climate  Important Questions for CBSE Class 7 Hindi Vasant Chapter 7 - Papa Kho Gaye  Important Questions for CBSE Class 8 Science Chapter 15 - Some Natural Phenomena  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 N  Previous Year Question Paper for CBSE Class 12 Physics - 2016 Set 1 N  CBSE Class 12 Maths Question Paper 2020  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  CBSE Class 10 Maths Question Paper 2017  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 - 2009  RD Sharma Class 12 Solutions Chapter 28 - Straight Line in Space (Ex 28.5) Exercise 28.5  RD Sharma Class 11 Solutions Chapter 28 - Introduction to 3D coordinate geometry (Ex 28.2) Exercise 28.2  RD Sharma Class 11 Solutions Chapter 28 - Introduction to 3D coordinate geometry (Ex 28.3) Exercise 28.3  RD Sharma Class 12 Solutions Chapter 28 - Straight Line in Space (Ex 28.1) Exercise 28.1  RD Sharma Class 12 Solutions Chapter 28 - Straight Line in Space (Ex 28.4) Exercise 28.4  RD Sharma Class 12 Solutions Chapter 28 - Straight Line in Space (Ex 28.3) Exercise 28.3  RD Sharma Class 12 Solutions Chapter 28 - Straight Line in Space (Ex 28.2) Exercise 28.2  RD Sharma Class 11 Solutions Chapter 28 - Introduction to 3D coordinate geometry (Ex 28.1) Exercise 28.1  RS Aggarwal Class 12 Solutions Chapter-28 The Plane  RS Aggarwal Class 11 Solutions Chapter-28 Differentiation  