Question

7 multiple choice 1 point
to simplify $x^{-6} \cdot x^{8}$ we must use the...
power of powers rule
both quotient and product rule
quotient of powers rule
both negative and quotient rule
8 multiple choice 1 point
simplify $x^{-6} \cdot x^{8}$

Explanation:

Step1: Identify exponent rule for product

When multiplying terms with the same base, add exponents: $x^a \cdot x^b = x^{a+b}$. Here, we have $x^{-6} \cdot x^8$, so we use the Product Rule. We can also rewrite the negative exponent using the definition $x^{-a}=\frac{1}{x^a}$, but the core operation to simplify the product relies on the Product Rule, and the negative exponent can be handled alongside it, meaning we use both the negative exponent definition and the product rule, which aligns with the option framing.

Step2: Match to correct option

The expression requires using the Product Rule (to combine the like bases) and handling the negative exponent, which corresponds to using both the negative exponent concept and the product rule, but among the given options, the only valid fit is recognizing that we use the product rule, and the option "Both Quotient and Product Rule" is incorrect, "Power of Powers" is for $(x^a)^b$, "Quotient Rule" is for $\frac{x^a}{x^b}$. The correct approach uses the Product Rule and the negative exponent rule, which is best matched to the idea that we use the product rule, and the option that fits is:

Wait, correcting: The key is $x^{-6} \cdot x^8 = x^{-6+8}=x^2$. This uses the Product Rule for Exponents (adding exponents when multiplying like bases) and we can also recognize that $x^{-6}=\frac{1}{x^6}$, but the primary rule to combine the terms is the Product Rule. Among the options, the only one that makes sense is that we use the Product Rule, and the option "Both Quotient and Product Rule" is wrong, but actually, no—wait, the negative exponent can be rewritten as a quotient, but when multiplying, we use the product rule. The correct option here is that we use the Product Rule, but since that's not listed, re-evaluating:

Wait, $x^{-6} \cdot x^8 = \frac{1}{x^6} \cdot x^8 = \frac{x^8}{x^6} = x^{8-6}=x^2$. Oh, this way we use the Quotient Rule (when we rewrite the negative exponent as a quotient and then subtract exponents) and the Product Rule (when we first combine the terms as a product). So this uses both the Quotient and Product Rule.

Answer:

Both Quotient and Product Rule