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If $12^{12x}\times4^{24x+12}\times5^{2y}=8^{4z}\times20^{12x}\times243^{3x-6}$, where $x$, $y$ and $z$ are natural numbers, then $x+y+z$ equals

If $9^{x^2+2x-3} - 4\left(3^{x^2+2x-2}\right) + 27 = 0$, then the product of all possible values of $x$ is

If $3^{a}=4,4^{b}=5,5^{c}=6,6^{d}=7,7^{e}=8$ and $8^{f}=9$, then the value of the product $\textit{abcdef }$ is

If $(x+6\sqrt{2})^{\frac{1}{2}} - (x-6\sqrt{2})^{\frac{1}{2}} = 2\sqrt{2}$, then x equals

The sum of all real values of $k$ for which $\small \left( \dfrac{1}{8} \right)^k \times \left( \dfrac{1}{32768} \right)^{\frac{1}{3}} = \left( \dfrac{1}{8} \right) \times \left( \dfrac{1}{32768} \right)^{\frac{1}{k}}$ is

Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of $2$. Then, the smallest possible value of $n + m$ is

The sum of all possible values of $x$ satisfying the equations $2^{4 x^{2}}-2^{2 x^{2}+x+16}+2^{2 x+30}=0$, is

Let $a$, $b$, $m$ and $n$ be natural numbers such that $a > 1$ and $b > 1$. If $a^m b^n = 144^{145}$, then the largest possible value of $n - m$ is

If$\sqrt{5x+9}+\sqrt{5x-9} = 3(2+\sqrt{2})$, then $\sqrt{10x + 9}$ is equal to

If $\left(\sqrt{\frac{7}{5}}\right)^{3 x-y}=\frac{875}{2401}$ and $\left(\frac{4 a}{b}\right)^{6 x-y}=\left(\frac{2 a}{b}\right)^{y-6 x}$

for all non-zero real values of $a$ and $b$, then the value of $x + y$ is

The number of integer solutions of the equation $\left(x^{2}-10\right)^{\left(x^{2}-3 x-10\right)}=1$ is

If $a, b, c$ are non-zero and $14^a = 36^b = 84^c$, then $6b \left( \frac{1}{c} - \frac{1}{a} \right)$ is equal to

The number of integers that satisfy the equality $(x^2 - 5x + 7)^{x + 1} = 1$ is

If $x=(4096)^{7+4 \sqrt{3}}$, then which of the following equals $64$ ?

How many distinct positive integer-valued solutions exist to the equation $\left(x^{2}-7 x+11\right)^{\left(x^{2}-13 x+42\right)}=1 ?$

If $5^{x}-3^{y}=13438$ and $5^{x-1}+3^{y+1}=9686$, then $x+y$ equals

If $(5.55)^x= (0.555)^y = 1000$, then the value of $\frac{1}{x} - \frac{1}{y}$ is

If $m$ and $n$ are integers such that $(\sqrt{2})^{19} 3^{4} 4^{2} 9^m8^n = 3^n 16^m (\sqrt[4]{64})$ then $m$ is

Given that $x^{2018}y^{2017} = 1/2$ and $x^{2016}y^{2019} = 8$, the value of $x^2 + y^3$ is

If $9^{x-\frac{1}{2}} - 2^{2x-2} = 4^x - 3^{2x-3}$, then x is

If $9^{2 x-1}-81^{x-1}=1944$, then $x$ is