First, we know that $\displaystyle 16$ is divisible by $\displaystyle 2$.

Therefore:

$\displaystyle 16=2^4$

The answer is:

$\displaystyle 2^4$

First, we know $\displaystyle 8$ is divisible by $\displaystyle 2$.

Therefore:

 $\displaystyle 8=2^3$

We can set-up an equation by applying the power rule of exponents.

$\displaystyle 2^{15}=(2^3)^x$ 

When a number to a power is raised by an exponent, we add the exponents. Write an equation and solve for $\displaystyle x$.

$\displaystyle 3x=15$

Simplify.

$\displaystyle x=5$

The answer is:

$\displaystyle 8^5$

First, know $\displaystyle 9$ is divisible by $\displaystyle 3$.

Therefore:

 $\displaystyle 9=3^2$

We can set-up an equation by applying the power rule of exponents.

$\displaystyle 3^{72}=(3^2)^x$ 

When a number to a power is raised by an exponent, we add the exponents. Write an equation and solve for $\displaystyle x$.

$\displaystyle 2x=72$

Simplify.

$\displaystyle x=36$

The answer is:

$\displaystyle 9^{36}$

First, we know the following:

 $\displaystyle 27=3^3$

In order to figure out the whole exponent, we can apply the power rule for exponents.

$\displaystyle 27^4=(3^3)^4=3^{12}$

We can expand $\displaystyle \frac{1}{5}^2$ to the following:

 $\displaystyle \frac{1}{5}*\frac{1}{5}$ 

The product is: 

$\displaystyle \frac{1}{25}$

We can expand $\displaystyle \frac{3}{4}^3$ to the following:

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

The product is: 

$\displaystyle \frac{27}{64}$

First, we know that $\displaystyle \frac{1}{32}$ is divisible by $\displaystyle \frac{1}{2}$.

The conversion becomes: 

$\displaystyle \frac{1}{2}^5$ 

First, we know that $\displaystyle \frac{1}{125}$ is divisible by $\displaystyle \frac{1}{5}$.

The conversion becomes: 

 $\displaystyle \frac{1}{5}^3$

We know that exponents raised to the negative power will generate fractions. 

We also know that $\displaystyle 8$ is divisible by $\displaystyle 2$.

$\displaystyle 8=2^3$ 

However, since this a fraction, we then have the following:

$\displaystyle \frac{1}{8}=2^{-3}$. 

If fractions are raised to a positive integer exponents, we know we will generate fractions; however, if a fraction is raised by a negative integer exponent, our answer will be whole number instead.

For example: 

$\displaystyle \frac{1}{2}^{-2}=\frac{1}{\frac{1}{2^2}}=\frac{1}{\frac{1}{4}}=4$

The following rule has been used in this scenario:

$\displaystyle x^{-a}=\frac{1}{x^a}$ 

In this formula, $\displaystyle a$ is a positive exponent raising the base $\displaystyle x$.

We know $\displaystyle 256=16^2$.

Since we are dealing with a integer and converting to fractional base, we know we need to have a negative exponent.

The answer is:

$\displaystyle \frac{1}{16}^{-2}$