Answer:
Proved (See Explanation)
Step-by-step explanation:
Show that 3ⁿ⁺⁴ - 3ⁿ is divisible by 16.
This is done as follows
From laws of indices;
aᵐ⁺ⁿ = aᵐ * aⁿ.
So, 3ⁿ⁺⁴ can be written as 3ⁿ * 3⁴.
becomes
Factorize
3ⁿ * 5
5(3ⁿ)
<em>The expression can not be further simplified.</em>
<em>However, we can conclude that when 3ⁿ⁺⁴ - 3ⁿ is divisible by 16, because 5(3ⁿ) is a natural whole number as long as n is a natural whole number.</em>
first we try to semplify as much as possible the fractions.
First we decompose the numbers
789 = 3 * 263
548 = 2 * 2 * 137 = 2^2
they have nothing in common, so the first fraction remains so
89 = 89 (prime number)
887 = 887 (prime number)
they have nothing in common, so the second one remains so
688 = 2 * 2 * 2 * 2 * 43 = 2^4 * 43
8724 = 2 * 2 * 3 * 727 = 2^2 * 3 * 727
they have 2^2 (=4) in common, so the numerator and the denominator can be devided by 4
so:
now we calculate the common denominator
take the scomposition:
548 = 2^2 * 137
887 = 887
2181 = 3 * 727
take every number one time
so the common denominator is 2^2 * 137 * 887 * 727 * 3 = 1'060'131'756
now calculate every single numerator: first we divide each denominator by the common denominator, then we moltiply the result with each numerator
first fraction: 789/548
1060131756 / 548 = 1934547
1934547 * 789 = 1526357583
second fraction: 89/887
1060131756 / 887 = 1195188
1195188 * 89 = 106371732
third one: 172/2181
1060131756 / 2181 = 486076
486076 * 172 = 83605072
now we can delete the common denominator
1526357583x - 106371732 = 83605072
solve it
1526357583x = 83605072 + 106371732
1526357583x = 189976804
x = 1526357583 / 189976804
decompose
1526357583 = 887 * 3 * 3 * 727 * 263
189976804 = 137 * 2 * 2 * 211 * 53 * 31
nothing in common
x = 1526357583 / 189976804
