When you have an exponent outside of parentheses, it means it is necessary to distribute it to all parts of whatever numbers or variables are inside the parentheses.

Distribute the exponent 4 to both the 3 and the 4 inside the parentheses

$\displaystyle \frac{3^{4}}{4^{4}}$

Now take 3 to the fourth power (or 3 four times) in the numerator and 4 to the fourth power (or 4 four times) in the denominator

$\displaystyle \frac{3\cdot 3\cdot 3\cdot 3}{4\cdot 4\cdot 4\cdot 4}$

$\displaystyle \frac{81}{256}$

This fraction does not reduce so this is the final answer.

An exponent outside of a parentheses needs to be distributed to all the numbers and variables in the parentheses. An exponent raised to an exponent should be multiplied.

$\displaystyle \left ( 4x^{3}y^{2} \right )^{4}$

$\displaystyle 4^{4}x^{3\cdot 4}y^{2\cdot 4}$

$\displaystyle 256x^{12}y^{8}$

An exponent outside of a parentheses means the entire quantity is being raised to that power. In other words, the quantity inside the parentheses is being multiplied by itself the number of times the outside exponent says.

$\displaystyle \left ( 9x^{2}+4y^{3} \right )^{2}=(9x^2+4y^3)(9x^2+4y^3)$

Recall that when like bases are being multiplied together their exponents are added.

$\displaystyle 9x^2\cdot 9x^2+9x^2\cdot 4y^3+4y^3\cdot 9x^2+4y^3\cdot 4y^3$

$\displaystyle 81x^4+36x^2y^3+36x^2y^3+16y^6$

$\displaystyle 81x^4+72x^2y^3+16y^6$

$\displaystyle \left ( 2x^5\right )^3=\left ( 2x^5\right )\left ( 2x^5\right )\left ( 2x^5\right )$

$\displaystyle 8x^{15}$

First simplify the expression inside the parentheses.

$\displaystyle \left ( \frac{x^2y^3z^{-2}}{x^{-2}z^4}\right )^{-3}$

$\displaystyle \left ( \frac{x^4y^3}{z^6}\right )^{-3}$

Then distribute the exponent.

$\displaystyle \frac{x^{-12}y^{-9}}{z^{-18}}$

Rearrange the expression so that there are no more negative exponents.

$\displaystyle \frac{z^{18}}{x^{12}y^{9}}$

When exponents are being raised by another exponent, we just multiply the powers.

$\displaystyle (10^8)^7=10^{8*7}=10^{56}$

When exponents are being raised by another exponent, we just multiply the powers.

$\displaystyle (-16^{-5})^{-9}=(-16)^{-5*-9}=-16^{45}$

When exponents are being raised by another exponent, we just multiply the powers.

$\displaystyle ((3^5)^7)^8=3^{5*7*8}=3^{280}$

When exponents are being raised by another exponent, we just multiply the powers.

$\displaystyle (24^{12})^{\frac{1}{18}}=24^{12*\frac{1}{18}}=24^{\frac{2}{3}}$ 

When we have fractional exponents, we convert like this:

$\displaystyle x^{\frac{a}b{}}=\sqrt[b]{x^a}$. $\displaystyle b$ is the index of the radical which is the denominator of the fractional exponent, $\displaystyle a$ is the power that will raise the base which is the numerator of the fractional exponent and $\displaystyle x$ is the base. 

$\displaystyle 24^{\frac{2}{3}}=\sqrt[3]{24^2}=\sqrt[3]{576}$ The perfect cube we can get is $\displaystyle 64$.

$\displaystyle \sqrt[3]{576}=\sqrt[3]{64*9}=4\sqrt[3]{9}$

When dealing exponents being raised by a power, we multiply the exponents and keep the base.

$\displaystyle (2^9)^4=2^{9*4}=2^{36}$