46. a. Open-Ended Write \( y^{6} \) as a product of two powers with the same base in four different ways. Use only positive exponents. b. Write \( y^{6} \) as a product of two powers with the same base in four different ways, using negative or zero exponents in each product. c. Reasoning How many ways can you write \( y^{6} \) as the product of two powers? Explain your reasoning.

To express \( y^{6} \) as a product of two powers with the same base, you can break it down in several ways. For positive exponents, you could write: 1. \( y^{3} \cdot y^{3} \) 2. \( y^{5} \cdot y^{1} \) 3. \( y^{4} \cdot y^{2} \) 4. \( y^{2} \cdot y^{4} \) For the case involving negative or zero exponents, you can creatively write: 1. \( y^{6} \cdot y^{0} \) 2. \( y^{7} \cdot y^{-1} \) 3. \( y^{8} \cdot y^{-2} \) 4. \( y^{10} \cdot y^{-4} \) When considering how many ways you can write \( y^{6} \) as the product of two powers, count not only the combinations of positive exponents but also the arrangements using negative and zero exponents. Each unique pair of integers that sum to 6 (and can be either positive, negative or zero) can be manipulated to create new forms. Thus, the combinations can be visually represented, leading to a variety of valid outputs. The total number is not fixed but rather depends on how creatively you can combine exponents that yield the sum of 6.