The problem requires simplifying ten different expressions involving exponents and fractions. The expressions cover various rules of exponents, including quotient of powers, power of a power, power of a product, and power of a quotient.

1. Problem Description

The problem requires simplifying ten different expressions involving exponents and fractions. The expressions cover various rules of exponents, including quotient of powers, power of a power, power of a product, and power of a quotient.

2. Solution Steps

1. $\frac{5^7}{5^8}$: Using the quotient of powers rule, $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{5^7}{5^8} = 5^{7-8} = 5^{-1} = \frac{1}{5}$.

2. $8^{24}$: There is no further simplification possible without knowing the value of $8^{24}$. So we leave it as $8^{24}$.

3. $(\frac{5}{9})^7$: Using the power of a quotient rule, $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Thus, $(\frac{5}{9})^7 = \frac{5^7}{9^7}$.

4. $\frac{8^4}{8^3}$: Using the quotient of powers rule, $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{8^4}{8^3} = 8^{4-3} = 8^1 = 8$.

5. $9^8 \cdot 9^2$: Using the product of powers rule, $a^m \cdot a^n = a^{m+n}$. Thus, $9^8 \cdot 9^2 = 9^{8+2} = 9^{10}$.

6. $\frac{8^2}{8^2}$: Any non-zero number divided by itself is

1. Thus, $\frac{8^2}{8^2} = 1$. Alternatively, using the quotient rule, $\frac{8^2}{8^2} = 8^{2-2} = 8^0 = 1$.

7. $(7 \cdot 8)^3$: Using the power of a product rule, $(ab)^n = a^n b^n$. Thus, $(7 \cdot 8)^3 = 7^3 \cdot 8^3$.

8. $8^{6^9}$: Here, $6^9$ is the exponent of

8. This expression remains as $8^{6^9}$. Note that this is different from $(8^6)^9$.

9. $(\frac{7}{5})^5$: Using the power of a quotient rule, $(\frac{a}{b})^n = \frac{a^n}{b^n}$. Thus, $(\frac{7}{5})^5 = \frac{7^5}{5^5}$.

1

0. $\frac{7^7}{7^9}$: Using the quotient of powers rule, $\frac{a^m}{a^n} = a^{m-n}$. Thus, $\frac{7^7}{7^9} = 7^{7-9} = 7^{-2} = \frac{1}{7^2} = \frac{1}{49}$.

3. Final Answer

1. $\frac{1}{5}$

2. $8^{24}$

3. $\frac{5^7}{9^7}$

4. $8$

5. $9^{10}$

6. $1$

7. $7^3 \cdot 8^3$

8. $8^{6^9}$

9. $\frac{7^5}{5^5}$

1

0. $\frac{1}{49}$