Brijesh Dwevedi and Jitendra Gupta Solutions for Chapter: Number Systems, Exercise 6: Challengers*

If $a^x=b, b^y=c$ and $xyz=1$, then what is the value of $c^z$?

If $2^m+2^{1+m}=96$, then find the minimum prime order pair for $m$ which satisfy the given equation

For a positive integer $n$, define $d\left(n\right)=$ nụmber of positive divisors of $n$. What is the value of $d\left[d\left(d\left(12\right)\right)\right]$?

If $\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1$, then $x$ is equal to

If $x={\left(4+\sqrt{15}\right)}^{1/3}+{\left(4-\sqrt{15}\right)}^{1/3}$, then $x^3-3x$ is ecual to

If $a^{1/3}+b^{1/3}+c^{1/3}=0$, then $\left(a+b+c\right){}^3$ is equal to

If $x=\frac{1}{2}\left(\sqrt{a}+\frac{1}{\sqrt{a}}\right)$, then $\frac{\sqrt{x^2}-1}{x-\sqrt{x^2}-1}$ is equal to

The simplified form of $\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$ is