Let's go through each problem in the image you uploaded and provide the solutions:

1. Simplify the division $\frac{2^{16}}{2^{14}}$

Using the property of exponents, $\frac{a^m}{a^n} = a^{m-n}$, we can simplify the expression:

$\frac{2^{16}}{2^{14}} = 2^{16-14} = 2^2 = 4$

So, the answer is $\boxed{4}$.

2. Simplify $3^7 \cdot 3^3 \cdot 3^{-6} \cdot 3^4$

Using the property of exponents, $a^m \cdot a^n = a^{m+n}$, we can combine all the terms:

$3^7 \cdot 3^3 \cdot 3^{-6} \cdot 3^4 = 3^{7+3+(-6)+4} = 3^8$

So, the answer is $\boxed{3^8}$.

3. Simplify $\sqrt[4]{2} \cdot \sqrt[4]{2^8} \cdot \sqrt[4]{2^{-12}}$

We convert the roots into exponent form. Recall that $\sqrt[n]{a^m} = a^{m/n}$:

[ \sqrt[4]{2} = 2^{\frac{1}{4}}, \quad \sqrt[4]{2^8} = 2^{\frac{8}{4}} = 2^2, \quad \sqrt[4]{2^{-12}} = 2^{\frac{-12}{4}} = 2^{-3} ]

Now, we combine the exponents using $a^m \cdot a^n = a^{m+n}$:

$2^{\frac{1}{4}} \cdot 2^2 \cdot 2^{-3} = 2^{\frac{1}{4} + 2 - 3} = 2^{\frac{1}{4} - 1} = 2^{-\frac{3}{4}}$

So, the answer is $\boxed{2^{-\frac{3}{4}}}$.

4. Calculate $4 \log 256$

Using the logarithmic property $a \log b = \log b^a$, we can simplify:

$4 \log 256 = \log 256^4$

Since $256 = 2^8$, we can rewrite this as:

$\log (2^8)^4 = \log 2^{32}$

Now using $\log a^b = b \log a$, we get:

$\log 2^{32} = 32 \log 2$

If the base of the logarithm is 2, then $\log_2 2 = 1$, so the answer is:

$\boxed{32}$

5. Simplify $\sqrt[4]{2^8} \cdot \sqrt{2}$

We convert the roots into exponent form. Recall that $\sqrt[n]{a^m} = a^{m/n}$ and $\sqrt{a} = a^{1/2}$:

$\sqrt[4]{2^8} = 2^{\frac{8}{4}} = 2^2, \quad \sqrt{2} = 2^{\frac{1}{2}}$

Now, we combine the exponents:

$2^2 \cdot 2^{\frac{1}{2}} = 2^{2 + \frac{1}{2}} = 2^{\frac{5}{2}}$

So, the answer is $\boxed{2^{\frac{5}{2}}}$.

6. Calculate $5 \log 5^7$

Using the logarithmic property $\log a^b = b \log a$, we simplify:

$5 \log 5^7 = 5 \cdot 7 \log 5 = 35 \log 5$

If the base of the logarithm is 5, then $\log_5 5 = 1$, so the answer is:

$\boxed{35}$

Would you like further details on any of the solutions? Here are some related questions to deepen understanding:

1. How do the properties of logarithms simplify complex expressions?
2. Can you provide more examples of simplifying expressions using exponent rules?
3. What happens when the logarithmic base is not provided, and how do you assume the base?
4. How would the result change in problem 5 if the base was a different number?
5. What other properties of logarithms could be applied in advanced problems?

Tip: Always remember that when dealing with exponents and logarithms, their properties make it easier to combine or break apart expressions.