[table][tr][td][math]2\cdot2\cdot2\cdot2[/math][/td][td][math]2^4[/math][/td][/tr][tr][td][math]16[/math][/td][td][math]4\cdot2[/math][/td][/tr][/table]

[math]2^4=2\cdot4[/math]

Explain how you know.

[math]3+3+3+3+3=3^5[/math]

Explain how you know.

[math]5^3=5\cdot5\cdot5[/math]

Explain how you know.

[math]2^3=3^2[/math]

Explain how you know.

[math]16^1=8^2[/math]

Explain how you know.

[math]\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=4\cdot\frac{1}{2}[/math]

Explain how you know.

[math]\left(\frac{1}{2}\right)^4=\frac{1}{8}[/math]

Explain how you know.

[math]8^2=4^3[/math]

Explain how you know.

[size=150]In the list, find expressions that are equivalent to each other and explain to your partner why they are equivalent. Your partner listens to your explanation. If you disagree, explain your reasoning until you agree. Switch roles for each list. (There may be more than two equivalent expressions in each list.)[/size][br][br][list][*][math]5\cdot5[/math][/*][*][math]2^5[/math][/*][*][math]5^2[/math][/*][*][math]2\cdot5[/math][/*][/list]

[list][*][math]4^3[/math][br][/*][*][math]3^4[/math][br][/*][*][math]4\cdot4\cdot4[/math][br][/*][*][math]4+4+4[/math] [/*][/list]

[list][*][math]6+6+6[/math] [/*][*][math]6^3[/math] [/*][*][math]3^6[/math] [/*][*][math]3\cdot6[/math][/*][/list]

[list][*][math]11^5[/math][br][/*][*][math]11\cdot11\cdot11\cdot11\cdot11[/math][br][/*][*][math]11\cdot5[/math][br][/*][*][math]5^{11}[/math][/*][/list]

[list][*][math]\frac{1}{5}\cdot\frac{1}{5}\cdot\frac{1}{5}[/math][br][/*][*][math]\left(\frac{1}{5}\right)^3[/math][br][/*][*][math]\frac{1}{15}[/math][br][/*][*][math]\frac{1}{125}[/math] [/*][/list]

[list][*][math]\left(\frac{5}{3}\right)^2[/math][br][/*][*][math]\left(\frac{3}{5}\right)^2[/math][br][/*][*][math]\frac{10}{6}[/math][br][/*][*][math]\frac{25}{9}[/math][/*][/list]

What is the last digit of [math]3^{1,000}[/math]? Show or explain your reasoning.