Question

Which number is greater \[{\left( {31} \right)^{12}}\] or ${\left( {17} \right)^{17}}$?

Answer 1

Hint: We will use inequalities simultaneously. First, we will use the relation 32 > 31 and after that we will raise the powers on both sides by 12. Similarly, 17 > 16 will be the next relation and again we will raise the powers by 17 both sides. We will convert the terms in the powers of 2 and then we will use the algebraic formula ${\left( {{a^m}} \right)^n} = {a^{mn}}$. Now, we will substitute the powers of 2 in any one of the equations and we will get the greater term.

Complete step-by-step answer:
We need to find the greater term between \[{\left( {31} \right)^{12}}\]and ${\left( {17} \right)^{17}}$.
Let us solve it by using the assumption of inequalities.
We know that 32 > 31.
We can write 32 as $2^5$. Putting this value in the above inequality, we get
$ \Rightarrow $$2^5$ > 31
Raising the power both sides by 12, we get
$ \Rightarrow $${\left( {{2^5}} \right)^{12}}$> ${\left( {31} \right)^{12}}$
We can write this equation by using the formula${\left( {{a^m}} \right)^n} = {a^{mn}}$as:
$ \Rightarrow {\left( 2 \right)^{60}}$> ${\left( {31} \right)^{12}}$ equation – (1)
Now, let us consider another inequality: 17 > 16
We can write 16 as $2^4$. Putting this value in the above inequality, we get
$ \Rightarrow $17 > $2^4$
Raising the power both sides by 17, we get
$ \Rightarrow $${\left( {17} \right)^{17}}$> ${\left( {{2^4}} \right)^{17}}$
We can write this equation by using the formula ${\left( {{a^m}} \right)^n} = {a^{mn}}$as:
$ \Rightarrow $${\left( {17} \right)^{17}}$> ${\left( 2 \right)^{68}}$ equation – (2)
Now, from observation, we can tell that ${\left( 2 \right)^{68}}$is greater than ${\left( 2 \right)^{60}}$i.e., ${\left( 2 \right)^{68}}$ > ${\left( 2 \right)^{60}}$.
Using the above relation and the upon combining equation (1) and (2), we get
$ \Rightarrow {\left( {17} \right)^{17}} > {\left( 2 \right)^{68}} > {\left( 2 \right)^{60}} > {\left( {31} \right)^{12}}$
Therefore, we have ${\left( {17} \right)^{17}}$> ${\left( {31} \right)^{12}}$.
Hence ${\left( {17} \right)^{17}}$is greater than ${\left( {31} \right)^{12}}$.

Note: In such questions, you may get caught up between the methods to solve this question. As you can solve this question by calculating the real values of the given terms by expanding the powers but it will be a very long and time taking procedure. Here, we assumed a number which is just greater than or equal to the number provided and there we established an inequality between them so as to make it a simple and quick way to reach the answer.