Why learn about roots and rational exponents?

Go to sleep!

Joan recently adopted a cat, Hobbes. He’s a great study buddy, but he has one small flaw: Hobbes keeps Joan up at night, asking to go out and come back in throughout the night as Joan is trying to sleep. So she wants to devise a way for Hobbes to get in and out of her apartment without her needing to open the door for him. After watching the following video on YouTube one day, she decides she will get her little sister Anne, skilled at Do-It-Yourself projects, to help her build a cat ladder so Hobbes will let her sleep at night.

Anne is pretty clever,  so Joan thinks she can help her with the complicated parts.

Joan’s window is located 12 feet above the ground, and the edge of the building is 5 feet from the edge of her window, so Joan needs to figure out how long of a ladder to make so it is confined within this space.

Even though Joan feels pretty confident about some of the things she has learned in her math class, she is not sure how to find the length for the ladder. She asks her math teacher for advice. Her teacher says that Joan can use the Pythagorean Theorem to find the length of wood she needs for her ladder.

Joan looks up the Pythagorean Theorem and discovers that it is an equation which describes the relationship between the lengths of the sides of a right triangle, such as the one a ladder will make with the side of her apartment building. Wikipedia says it looks like this:

[latex]{a}^{2}+{b}^{2}={c}^{2}[/latex]

Joan learns that [latex]c^2[/latex] represents the long side of the right triangle, so in her case this is the length of ladder that she will need. She gets to work trying to solve the equation:

[latex]\begin{array}{ccc}\hfill {a}^{2}+{b}^{2}& =& {c}^{2}\hfill \\ \hfill {5}^{2}+{12}^{2}& =& {c}^{2}\hfill \\ \hfill 169& =& {c}^{2}\hfill \end{array}[/latex]

Joan gets this far and realizes she does not understand how to solve this kind of equation. She needs to find out the length that, when squared, is 169, to determine the length of wood she needs.

What she doesn’t realize is that she needs to find a square root.

In this module, we will investigate methods of finding solutions to problems such as this one.  We will revisit Joan later and see if she and Anne were successful in making a cat ladder for Hobbes.