Exponents and Surds

Prove that, ${\left(a^{\frac{1}{x-y}}\right)}^{\frac{1}{x-z}}\times {\left(a^{\frac{1}{y-z}}\right)}^{\frac{1}{y-x}}\times {\left(a^{\frac{1}{z-x}}\right)}^{\frac{1}{z-y}}=1$

Solve for $x:$

$3^x=2^{-x}$

If $x+z=2y$ and $b^2=ac$, then let us show that, $a^{y-z}{b}^{z-x}{c}^{x-y}=1$

If $a^x=b^y=c^z$ and $b^2=ac$, then find the value of $y$.

If ${\left(\frac{9}{4}\right)}^x{\left(\frac{8}{27}\right)}^{x-1}=\frac{2}{3}$, then find value of $x$.

If $abc=1$, then what is the value of $\left(\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}\right)$

Find the value of ${\left(\frac{x^a}{x^b}\right)}^{\frac{1}{ab}}{\left(\frac{x^b}{x^c}\right)}^{\frac{1}{bc}}{\left(\frac{x^c}{x^a}\right)}^{\frac{1}{ca}}$

Find the value of $\frac{{\left(243\right)}^{{\displaystyle \frac{n}{5}}}\times 3^{2n+1}}{9^n\times 3^{n-1}}$

Is the product of the square root of $x$ with the cube root of $x$, is sixth root of the fifth power of $x$. (Yes $/$ No)

Write ${\left(\frac{1}{9}\right)}^{\frac{-1}{2}}\times {\left(64\right)}^{\frac{-1}{3}}$ as a rational number.

If $a$ and $b$ are distinct positive primes such that $\sqrt[3]{a^6{b}^{-4}}=a^x{b}^{2y}$, then find the value of $x+y$, as an improper fraction.

Assuming that $x$ is a positive real numbers, if the simplified form of ${\left(\sqrt{x^{-3}}\right)}^5$ is $\frac{1}{x^{\frac{m}{n}}}$, then find the value of $m+n$.

Find the value of $\sqrt{248+\sqrt{51+\sqrt{169}}}$.

If $\sqrt{2401}=\sqrt{7^x}$, then find the value of $x$.

Find the value of $\sqrt{5+\sqrt{11+\sqrt{18+\sqrt{49}}}}$.

The area of a square field is $60025 {\mathrm{m}}^2$. A boy cycles along its boundary at $18 \mathrm{km}/h$. In how much time will he return to the starting point?

The square root of a quotient is equal to the product of the square roots of the _____ and divisor.

Simplify $\sqrt{\frac{48}{81}}$.

Simplify $\frac{\sqrt{12}}{\sqrt{3}}$

The ${100}^{th}$ root of ${10}^{\left({10}^{10}\right)}$ is