1. We multiplied: x to the exponent of 3 muliplied by x to the exponent of 4.

a) Suppose the exponents 3 and 4 were different. Would the adding method still work? Explain.

b) Suppose the base x was different. Would we get similar results? Explain.

2. We divided: x to the exponent of 5 divided by x to the exponent of 3.

a) Suppose the exponents 5 and 3 were different. Would the subraction method still work? Explain.

b) Suppose the base x was different. Would we get similar results? Explain.

1. x^3 * x^4 so use the base x and add the exponents.
a) yes, because you're multiplying a common base.
b) No, if the bases were different we can't add the exponents.
2. x^5 / x^3
a) yes, the base is the same
b) No, the results would be different.

To explain how we arrive at the answers, let's break down the steps:

1. For multiplying powers with the same base, like x^3 * x^4, we add the exponents: x^(3+4) = x^7. This is known as the "Product of Powers" rule.

a) If the exponents were different, let's say x^3 * x^5, we would still use the same method of multiplying the common base, but the result would be x^(3+5) = x^8. So yes, the adding method would still work regardless of whether the exponents are different, as long as the bases are the same.

b) However, if the base x was different, like y^3 * x^4, we cannot directly add the exponents because the bases are not the same. In this case, we cannot simplify the expression further using the exponent rules.

2. For dividing powers with the same base, like x^5 / x^3, we subtract the exponents: x^(5-3) = x^2. This is known as the "Quotient of Powers" rule.

a) If the exponents were different, like x^5 / x^4, we would still use the subtraction method: x^(5-4) = x^1 = x. So yes, the subtraction method would still work regardless of whether the exponents are different, as long as the bases are the same.

b) Different base results in different results. If the base x was different, like y^5 / x^3, we cannot simplify the expression any further using the exponent rules because the bases are not the same. So the results would be different.

In summary, the rules for multiplying and dividing powers depend on whether the bases are the same or different, and whether the exponents are the same or different.