Let's go through each problem one by one and solve them.

Problem 22:

$8^6 \cdot 8^5 \cdot 8^2$ Since the bases are the same, we can use the property of exponents: $a^m \cdot a^n = a^{m+n}$. Thus: $8^{6 + 5 + 2} = 8^{13}$

Problem 23:

$(16 \cdot 40)^{\frac{3}{8}}$ First, calculate $16 \cdot 40 = 640$, so the expression becomes: $640^{\frac{3}{8}}$ Now, express 640 as a power of 2: $640 = 2^7 \cdot 5$. Approximating further without exact simplification leads to a more involved calculation, but for now, we can express it as $(640)^{3/8}$.

Problem 24:

$(3^4)^{\frac{1}{3}}$ Using the power of a power rule $(a^m)^n = a^{m \cdot n}$: $3^{4 \cdot \frac{1}{3}} = 3^{\frac{4}{3}}$

Problem 25:

$\frac{32^{19/20}}{32^{3/4}}$ Use the quotient rule of exponents $\frac{a^m}{a^n} = a^{m-n}$: $32^{19/20 - 3/4} = 32^{\frac{19}{20} - \frac{15}{20}} = 32^{\frac{4}{20}} = 32^{\frac{1}{5}}$

Problem 26:

$(49 \cdot 9)^{-\frac{1}{2}}$ First, calculate $49 \cdot 9 = 441$. So the expression becomes: $441^{-\frac{1}{2}} = \frac{1}{\sqrt{441}} = \frac{1}{21}$

Problem 27:

$\left(\frac{625}{16}\right)^{\frac{3}{4}}$ We can break this into two parts: $625^{\frac{3}{4}} \quad \text{and} \quad 16^{\frac{3}{4}}$ We know $625 = 5^4$, so: $625^{\frac{3}{4}} = 5^3 = 125$ And $16 = 2^4$, so: $16^{\frac{3}{4}} = 2^3 = 8$ Thus, the final expression is: $\frac{125}{8}$

If you want more details or have any specific questions, feel free to ask!

Here are 5 related questions:

1. What is the rule for multiplying powers with the same base?
2. How do you simplify a power raised to another power?
3. How do you convert fractional exponents into radicals?
4. How can you express numbers as powers of prime factors to simplify exponent calculations?
5. What are the properties of negative exponents?

Tip: Always simplify the base before applying the exponent rules; this can make calculations easier!