Question

How do I simplify? $ \dfrac{{{5^n} + {5^{n + 1}}}}{{{5^{n + 1}} + {5^n}}} $

Answer 1

Hint: : In order to simplify the above equation, split the term $ {5^{n + 1}} $ using the law of radicals, then take out the common values if present. Cancel the terms if common and solve the terms as needed and get the required answer.

Complete step by step solution:
We are given with an equation $ \dfrac{{{5^n} + {5^{n + 1}}}}{{{5^{n + 1}} + {5^n}}} $ .
From the law of radical’s, we know that in multiplication, if the base is common then their powers would be added as: $ {p^a}.{p^b} = {p^{a + b}} $ .Similarly, this property can be used in splitting the value and for $ {5^{n + 1}} $ , it can be written as $ {5^{n + 1}} = {5^n}{.5^1} $ .
Substituting this value in our equation, we get:
 $ \dfrac{{{5^n} + {5^{n + 1}}}}{{{5^{n + 1}} + {5^n}}} = \dfrac{{{5^n} + {5^n}.5}}{{{5^n}.5 + {5^n}}} $
We can see that $ {5^n} $ , is common in both the denominator and numerator, so taking out the common we get:
 $ \dfrac{{{5^n} + {5^n}.5}}{{{5^n}.5 + {5^n}}} = \dfrac{{{5^n}\left( {1 + 5} \right)}}{{{5^n}\left( {5 + 1} \right)}} $
Cancelling $ {5^n} $ , from the numerator and denominator and we are left with:
 $ \dfrac{{{5^n}\left( {1 + 5} \right)}}{{{5^n}\left( {5 + 1} \right)}} = \dfrac{{1 + 5}}{{5 + 1}} $
Further adding the terms:
 $ \dfrac{{1 + 5}}{{5 + 1}} = \dfrac{6}{6} $
Cancelling/dividing the terms and we get:
 $ \dfrac{6}{6} = 1 $
Therefore, $ \dfrac{{{5^n} + {5^{n + 1}}}}{{{5^{n + 1}} + {5^n}}} = 1 $ .
So, the correct answer is “1”.

Note: When the base would be the same during multiplication then the powers would be added to the variables and would be stored in any one variable present. For ex: $ {p^a}.{p^b} = {p^{a + b}} $
When the base would be the same during division of variables, then powers would be subtracted. For Ex: $ \dfrac{{{p^a}}}{{{p^b}}} = {p^{a - b}} $