How can I use the properties of integer exponents to simplify algebraic abd numeric expressions?

You have to follow the laws of exponents. For exponents as integers, the law that are applicable for simplification are the following:

1. If the two algebraic or numeric terms has the same base, you can add their exponents.

Ex: a² + a³ = a²⁺³ = a⁵ or 6² + 6⁵ = 6⁷

2. When an exponent is outside a base raised to another exponent, simply multiple the exponents:

Ex: (a²) ³ = a⁶

3. If you divide two terms with the same bases, just subtract the exponent of the numerator to the denominator.

Ex: a³/a² = a³⁻² = a¹ or a

4. When any base is raised to the power of zero, the answer is 1.

Ex: 100,000⁰ = 1

5. If the integer exponent is negative, take the reciprocal to make the exponent positive.

Ex: a⁻³ = 1/a³