Laws of Exponents

Simplify: ${\left(ab\right)}^{y-z}.{\left(bc\right)}^{z-x}.{\left(ca\right)}^{x-y}$

Simplify: $\frac{5^{x+2}-6\times 5^{x+1}}{13\times 5^x-2\times 5^{x+1}}$

If $x+z = 2y$ and $b^2=ac$, then prove that $a^{\left(y-z\right)}\times b^{\left(z-x\right)}\times c^{\left(x-y\right)}=1$

If $5^{2x-1}-{25}^{x-1}=2500$ then find the value of $x$

Evaluate:

${\left(xy\right)}^{m-n}.{\left(yz\right)}^{n-l}.{\left(zx\right)}^{l-m}$

Solve: $\frac{3\times {27}^{n+1}+9\times 3^{3n-1}}{8\times 3^{3n}-5\times {27}^n}$

Solve: ${\left(27x^{-3}{y}^6\right)}^{\frac{2}{3}}$

Simplify ${\left(36x^2\right)}^{\frac{1}{2}}$

Simplify: ${\left(a^{-2}b\right)}^{-2}{\left(ab\right)}^{-3}$

Evaluate: $\frac{a^{2n+1}\times a^{\left(2n+1\right)\left(2n-1\right)}}{a^{n\left(4n-1\right)}\times {\left(a^2\right)}^{2n+3}}$

Find the value of $n$, when 

$\frac{a^{2n-3}\times {\left(a^2\right)}^{n+1}}{{\left(a^4\right)}^{-3}}={\left(a^3\right)}^3÷{\left(a^6\right)}^{-3}$

Simplify:

$\frac{3^{n+1}\times 9-{\displaystyle \frac{3}{{27}^{-n}}}}{3^{1+3n}}$

Find the value of $x$ in ${\left(18\right)}^{3.5}÷{\left(27\right)}^{3.5}\times 6^{3.5}=2^x$.

Find the value of $x$ in $3^{3.5}\times {21}^2\times {42}^{2.5}÷2^{2.5}\times 7^{3.5}={21}^x$.

Find the value of $x$ for each of the following.

${\left(\frac{-1}{3}\right)}^4÷{\left(\frac{-1}{3}\right)}^5={\left(\frac{-1}{3}\right)}^x$

Simplifying $2^5\times 3^5$, we get, $k^5$. Find $k$.

Using law of exponents, If the simplest form of ${\left(7^3\right)}^2$ is $7^k$, find $k$.

Using law of exponents, If the simplest form of  $4^8÷4^3$ is $4^k$, find $k$.

Using law of exponents, The simplest form of $2^4\times 2^5\times 2^3\times 2$ is $2^k$, find $k$.

Solve for $x$:

$\frac{{10}^x}{{10}^{-3}}={10}^9$