Question

How do you write a convincing argument to show why \[3^\circ = 1\] using the following pattern: \[{3^5} = 243,{3^4} = 81,{3^3} = 27,{3^2} = 9\] ?

Answer 1

Hint: The Quotient of Powers Property states that when dividing two exponents with the same base, you can subtract the exponents and keep the base. The power rule tells us that to raise a power to a power, just multiply the exponents and the quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. The quotient of two numbers is the result of the division of these numbers.

Complete step-by-step answer:
Given,
 \[3^\circ = 1\] ,
In which,
 \[{x^0} = 1\forall x \in R\] , which belongs to all Real numbers.
Hence, here we have the given sequence as \[{3^5},{3^4},{3^3},{3^2},..\] , hence according to this we have:
 \[{a_n} = {3^n}\] for \[n = 5 \to 2\]
We need to extend this to:
 \[{a_1} = {3^1} = 3\]
Hence, we have:
 \[ \Rightarrow \dfrac{{{a_{n - 1}}}}{{{a_n}}} = \dfrac{1}{3}\]
 \[ \Rightarrow {a_{n - 1}} = \dfrac{{{a_n}}}{3}\]
Now, let us consider:
 \[{a_0} = \dfrac{{{a_1}}}{3}\]
 \[ \Rightarrow {a_0} = \dfrac{3}{3} = 1\]
Since, \[{a_0} = 3^\circ \] the proposition is proved.

Note: We must know that when dividing two powers with the same base, we subtract the exponents. We must also know the Quotient to a Power law, as the fraction of two different bases with the same power is represented as; \[\dfrac{{{a_n}}}{{{b_n}}} = {\left( {\dfrac{a}{b}} \right)^n}\] , where a and b are non-zero terms and n is an integer.