Question

If ${x^a} = y$,\[{y^b} = z\]and ${z^c} = x$, then prove that $abc = 1$

Answer 1

Hint: We have to prove $abc = 1$ and for this first put the value of $y$ in ${y^b}$, then put value of $z$ in ${z^c}$ and finally by applying the concept of equality of a power we can prove the given statement. The concept of equality says that if the base of two numbers are same/equal then their powers are also equal. i.e.
If ${x^a} = {x^b}$ then $a = b$.

Complete step-by-step answer:
First of all, let us see what is given to us? We are given with the values of some variables i.e.
$ \Rightarrow {x^a} = y$, …………(1)
$ \Rightarrow {y^b} = z$ and ………….(2)
$ \Rightarrow {z^c} = x$ ……………(3)
Now let us see what we have to prove? We have to prove $abc = 1$.
To prove this firstly put the value of $y$ from (1) in equation (2), we get,
 $ \Rightarrow {\left( {{x^a}} \right)^b} = z$
 Here in the above equation you can see, $x$ has power a and they are in bracket outside the bracket there is also a power i.e. b. So, both the powers will get multiplied i.e. a and b are multiplied and we get,
 $ \Rightarrow {x^{ab}} = z$
 Now, put this value of z in equation (3), we get,
 $ \Rightarrow {\left( {{x^{ab}}} \right)^c} = x$
 Here in the above equation we can see, $x$ has power ab and they are in brackets outside the bracket there is also a power i.e. c. So, both the powers will get multiplies i.e. ab and c are multiplied and we get,
 $ \Rightarrow {x^{ab}}^c = x$
 In the above equation you can see that the base on both sides is the same but the powers are different. If the base of two terms is equal then their respective powers are also equal. Hence, we get,
 $ \Rightarrow abc = 1$
Hence, Proved.

Additional notes:
Some properties of powers are as follows:
1.When two numbers in powers are multiplied and having same base then it is given by
$ \Rightarrow {x^a} \times {x^b} = {x^{a + b}}$
2.When two numbers in powers are divided and having same base then it is given by
$ \Rightarrow \dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
3.Any number have power 0 is equal to 1 i.e.
$ \Rightarrow {x^0} = 1$

Note: In this question students can make mistakes while solving ${\left( {{x^a}} \right)^b}$, in this they may add the powers as they consider as the property of addition of powers and they can write it as ${x^{a + b}}$. This way they are enabled to reach the result. The property of addition is applied if the powers are on another base i.e. ${x^a}{x^b}$. But ${\left( {{x^a}} \right)^b}$ is in the form of double power of $x$ and hence multiplied and we get ${x^a}^b$.