Based on the power rule for exponents we can write:

$\displaystyle (a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. So we can write:

$\displaystyle \frac{3x^{\sqrt{2}}(x^{\sqrt{3}})^\sqrt{3}}{x^{\sqrt{2}}}=3\times \left(\frac{x^{\sqrt{2}}}{x^{\sqrt{2}}}\right) \times (x^{\sqrt{3}})^{\sqrt{3}}=3x^{\sqrt{3}\times \sqrt{3}}=3x^3$

Substitute $\displaystyle x=2$ and we get:

$\displaystyle 3x^3=3\times 2^3=3\times 8=24$

Based on the power rule, we know that in order to raise a power to a power we need to multiply the exponents, i.e.

$\displaystyle (x^{m})^{n}=x^{m\times n}$.

$\displaystyle (x^{2\sqrt{2}})^{3\sqrt{2}}=x^{2\sqrt{2}\times 3\sqrt{2}}=x^{2\times 3\times 2}=x^{12}$

The Negative Exponent Rule says $\displaystyle a^{-n}=\frac{1}{a^n}$.

$\displaystyle (\frac{12x^{3}}{3x^{-1}})^{4}=(\frac{12}{3}\times x^{3+1})^{4}=(4x^4)^{4}$

The power rule says that, in order to raise a power to a power, we need to multiply the exponents, i.e. $\displaystyle (x^{m})^{n}=x^{m\times n}$.

$\displaystyle (\frac{12x^{3}}{3x^{-1}})^{4}=(4x^4)^{4}=4^4\times x^{4\times 4}=256x^{16}$

When an exponent is raised to another exponent, the exponents should be multiplied toghether. This will result in:

$\displaystyle (x^{2})^{3}-(2^{3})^{2}$

$\displaystyle x^{6}-2^{6}$

$\displaystyle x^{6}-64$

When exponent of a value is raised to another exponent, the values of the exponents are multiplied by each other. 

$\displaystyle (x^{2})^{3}$

2 is multiplied by 3, and so the exponent of x is 6. 

$\displaystyle x^{6}$

When one exponent is raised to another exponent, the values of the exponents should be multiplied together. Thus, 

$\displaystyle (2^{6})^{\frac{1}{2}}$ can be simplified to $\displaystyle 2^{3}$, given that $\displaystyle 6\cdot \frac{1}{2}=3$. 

$\displaystyle 2^{3}=2\cdot2\cdot2=8$

When exponents are multiplied by each other, the powers should be added together. Meanwhile, numbers not raised to an exponent are simply multiplied by each other. 

Therefore, the answer is $\displaystyle 15x^{13}$, because $\displaystyle 3*5=15$, and $\displaystyle 9+4=13$. 

1 raised to any exponent will always be 1. 

-1 will be equal to 1 when the exponent is even and will be equal to -1 when the exponent is odd. 

Given that 323 is odd, $\displaystyle -1^{323}$ is equal to -1. 

Simplify the following:

$\displaystyle 6x^5*5x^6*3x^2$

Let's begin by recalling two rules

1) When multiplying variables with a common base, add the exponents.

2) When multiplying variables with a common base, multiply the coefficients.

$\displaystyle 6x^5*5x^6*3x^2=(5*6*3)x^{5+6+2}=90x^{13}$

So, our answer is 

$\displaystyle 90x^{13}$

Separate the fraction and apply the quotient of powers rule:

$\displaystyle \frac{12x^{12}}{3x^{3}} = \frac{12}{3} \cdot \frac{x^{12}}{x^{3}} = 4x^{12-3} = 4x^{9}$