We are asked to simplify ten expressions involving exponents and fractions.

1. Problem Description

We are asked to simplify ten expressions involving exponents and fractions.

2. Solution Steps

1. $\frac{5^7}{5^8}$:

Using the quotient rule, $\frac{a^m}{a^n} = a^{m-n}$.

$\frac{5^7}{5^8} = 5^{7-8} = 5^{-1} = \frac{1}{5}$.

2. $8^{2^4}$:

First, calculate the exponent $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$.

$8^{2^4} = 8^{16}$.

3. $(\frac{5}{9})^7$:

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

$(\frac{5}{9})^7 = \frac{5^7}{9^7}$.

4. $\frac{8^4}{8^3}$:

Using the quotient rule, $\frac{a^m}{a^n} = a^{m-n}$.

$\frac{8^4}{8^3} = 8^{4-3} = 8^1 = 8$.

5. $9^8 \cdot 9^2$:

Using the product rule, $a^m \cdot a^n = a^{m+n}$.

$9^8 \cdot 9^2 = 9^{8+2} = 9^{10}$.

6. $\frac{8^2}{8^2}$:

Any number divided by itself is

1. $\frac{8^2}{8^2} = 1$.

7. $(7 \cdot 8)^3$:

Using the power of a product rule, $(ab)^n = a^n b^n$.

$(7 \cdot 8)^3 = 7^3 \cdot 8^3$.

8. $8^{6^9}$:

This means $8$ raised to the power of $6^9$. We can rewrite this as $8^{(6^9)}$.

9. $(\frac{7}{5})^5$:

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

$(\frac{7}{5})^5 = \frac{7^5}{5^5}$.

1

0. $\frac{7^7}{7^9}$:

Using the quotient rule, $\frac{a^m}{a^n} = a^{m-n}$.

$\frac{7^7}{7^9} = 7^{7-9} = 7^{-2} = \frac{1}{7^2} = \frac{1}{49}$.

3. Final Answer

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

2. $8^{16}$

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. $\frac{1}{49}$