We need to simplify the expressions involving exponents. Problem 3: $m^{-1} = ?$ Problem 4: $9^0 = ?$ Problem 5: $7^6 \cdot 7^4 = ?$ Problem 6: $(y^{10})^5 = ?$ Problem 7: $\frac{8^{82}}{8^9} = ?$

1. Problem Description

We need to simplify the expressions involving exponents.

Problem 3: $m^{-1} = ?$

Problem 4: $9^0 = ?$

Problem 5: $7^6 \cdot 7^4 = ?$

Problem 6: $(y^{10})^5 = ?$

Problem 7: $\frac{8^{82}}{8^9} = ?$

2. Solution Steps

Problem 3: A negative exponent means taking the reciprocal of the base.

$m^{-1} = \frac{1}{m}$

Problem 4: Any number (except 0) raised to the power of 0 is

1. $9^0 = 1$

Problem 5: When multiplying exponential terms with the same base, add the exponents.

$a^m \cdot a^n = a^{m+n}$

$7^6 \cdot 7^4 = 7^{6+4} = 7^{10}$

Problem 6: When raising a power to another power, multiply the exponents.

$(a^m)^n = a^{m \cdot n}$

$(y^{10})^5 = y^{10 \cdot 5} = y^{50}$

Problem 7: When dividing exponential terms with the same base, subtract the exponents.

$\frac{a^m}{a^n} = a^{m-n}$

$\frac{8^{82}}{8^9} = 8^{82-9} = 8^{73}$

3. Final Answer

Problem 3: $\frac{1}{m}$

Problem 4: $1$

Problem 5: $7^{10}$

Problem 6: $y^{50}$

Problem 7: $8^{73}$