Power Up!

     Professor Lazanu is a world-class expert on exponents. He often enjoys showing his three precocious young nieces interesting and curious number facts and relationships based on the idea of bases and exponents.

     For example, one time he showed them the following equation:

a^b - b^a = 0

     They quickly solved that as 4² - 2⁴ = 0, because each power is 16.

     This inspired the girls to invent their own "power" equations to challenge their uncle.

     Soon Sheila shyly showed this one: a^b - b^a = 118

     Next Sylvia slyly stated: a^b - b^a = 7.

     Lastly Priscilla proudly pronounced: a^b - b^a = 1844.

     Not to be outdone by his nieces, he did so by stating the products of the a-value and the b-value for each case. What did the wily old professor say?

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     Extra: As Mrs. Lazanu brought in some milk and cookies for everybody, she offered this little teaser: "When does a^b and b^a represent consecutive squares?"