1. **State the problem:** Determine whether the expression $$\frac{2^{4m}}{16}$$ is equivalent to each of the given expressions: $$2^{4m-4}$$, $$4^{2m} \cdot 2$$, $$16^m \cdot 1$$. 2. **Rewrite the denominator:** Note that $$16 = 2^4$$, so the expression becomes: $$\frac{2^{4m}}{2^4}$$ 3. **Apply the quotient rule of exponents:** $$\frac{a^x}{a^y} = a^{x-y}$$ 4. **Simplify the expression:** $$\frac{2^{4m}}{2^4} = 2^{4m-4}$$ 5. **Check equivalence with each expression:** - Compare with $$2^{4m-4}$$: They are exactly the same, so **Equivalent**. - Compare with $$4^{2m} \cdot 2$$: Rewrite $$4^{2m}$$ as $$(2^2)^{2m} = 2^{4m}$$, so: $$4^{2m} \cdot 2 = 2^{4m} \cdot 2 = 2^{4m+1}$$ This is not equal to $$2^{4m-4}$$, so **Not equivalent**. - Compare with $$16^m \cdot 1$$: Rewrite $$16^m$$ as $$(2^4)^m = 2^{4m}$$, so: $$16^m \cdot 1 = 2^{4m}$$ This is not equal to $$2^{4m-4}$$, so **Not equivalent**. **Final answer:** - $$\frac{2^{4m}}{16} = 2^{4m-4}$$ is equivalent to $$2^{4m-4}$$ only. - It is not equivalent to $$4^{2m} \cdot 2$$ or $$16^m \cdot 1$$.