Question

read & understand (highlight or circle important keywords.)
naomi and douglas each have a baseball card collection. naomi has two thousand baseball cards in her collection. douglas has $(4^{2})^{3}$ baseball cards in his collection. who has more baseball cards?
what do we know? (list the important facts needed for this problem to be solved.)
what do we want to find out? (explain in your own words.)
solve (answer the problem. show all work.)
naomi → 2000
dooglas → $(4^{2})^{3}$
$(4^{2})^{3}=4^{2(3)}=4^{6}=$
explain (state your answer. give details. explain what you did and justify your steps.)

Explanation:

Step1: Simplify Douglas's card number

Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), for \((4^{2})^{3}\), we have \(m = 2\), \(n=3\), so \((4^{2})^{3}=4^{2\times3}=4^{6}\).
Then calculate \(4^{6}=4\times4\times4\times4\times4\times4=(4\times4)\times(4\times4)\times(4\times4)=16\times16\times16 = 4096\).

Step2: Compare the number of cards

Naomi has \(2000\) cards. Douglas has \(4096\) cards.
Since \(4096>2000\).

Answer:

Douglas has more baseball cards. Because Naomi has \(2000\) baseball cards and Douglas has \((4^{2})^{3}=4^{6}=4096\) baseball cards, and \(4096>2000\).