Noah is very proud of his sobriety. He credits some of his success to handicrafts, like beading. He finds the steady, repetitive work of stringing beads one by one (or, actually, Noah prefers two by two) to be a calming practice and he enjoys flexing his artistic creativity.
Noah’s mom also enjoys handicrafts and would like to build a small glass box to hold some of Noah’s beads. To keep things simple she has decided that the box will be a cube, meaning it will have the same length, width, and height. Plus, the cube has long been the symbol of regeneration and stability but also of limitations and boundaries – a fitting recovery gift.
She could make a small box that’s \(2 \times 2 \times 2\text{.}\) That’s pronounced “2 by 2 by 2” and means 2 inches long, 2 inches wide, and 2 inches tall. Such a small box would hold \(2 \times 2 \times 2 = 8\) cubic inches of beads, which is around 2/3rd of a cup (not much).
What if Noah’s mom made a box that was \(5 \times 5 \times 5\) instead? That box would hold \(5 \times 5 \times 5 = 125\) cubic inches of beads, which is just over 2 liters of beads. Similarly a \(10 \times 10 \times 10\) box would hold \(10 \times 10 \times 10 = 1000\) cubic inches of beads, or about 4 gallons of beads. In case you’re curious about the units here, there’s about 14.4 cubic inches per cup, 61 cubic inches per liter, and 231 cubic inches per gallon.
When we multiply a number by itself, like
\begin{equation*}
\underbrace{2 \times 2 \times 2}_{3 \text{ times}}
\end{equation*}
we say that we are raising 2 to the power of 3. The number 3, which counts how many times we multiply 2 by itself, is called the power or exponent.
On a calculator we can use the \(\wedge\) key. Try for yourself:
\begin{equation*}
2 \wedge 3 = \underbrace{2 \times 2 \times 2}_{3 \text{ times}} = 8
\end{equation*}
\begin{equation*}
5 \wedge 3 = \underbrace{5 \times 5 \times 5}_{3 \text{ times}} = 125
\end{equation*}
\begin{equation*}
10 \wedge 3 = \underbrace{10 \times 10 \times 10}_{3 \text{ times}} = 1000
\end{equation*}
No key marked \(\wedge\text{?}\) Look for a key marked \(x^y\) or \(y^x\) instead. Can’t find that either? Ask a classmate or your instructor for help.
We can even do higher powers for practice:
\begin{equation*}
3 \wedge 4 = \underbrace{3 \times 3 \times 3 \times 3}_{4 \text{ times}} = 81
\end{equation*}
\begin{equation*}
2 \wedge 10 = \underbrace{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}_{10 \text{ times}}= 1024
\end{equation*}
Speaking of which, Noah thought it was fitting it is that his mom was using mathematical higher powers to build the glass box since learning to trust in a spiritual higher power was so important to his recovery.
Noah would like the box to hold 1 gallon of beads which is about 231 cubic inches. What size cube box should his mom make? She wants a number, let’s call it \(n\text{,}\) so that an \(n \times n \times n\) box will hold 231 cubic inches. That means she needs
\begin{equation*}
n \times n \times n = 231
\end{equation*}
or, equivalently,
\begin{equation*}
n \wedge 3 = 231
\end{equation*}
Noah’s mom can try to guess what number \(n\) should be. She wants a box that’s bigger than \(5 \times 5 \times 5\) but smaller than \(10 \times 10 \times 10\text{.}\) Her first guess is \(n=7\) inches, meaning the box will be \(7 \times 7 \times 7\text{.}\) That box would hold
\begin{equation*}
7 \times 7 \times 7 = 7 \wedge 3 = 343 \text{ cubic inches}
\end{equation*}
which is too big. How about \(n = 6.5\) inches so the box would hold
\begin{equation*}
6.5 \times 6.5 \times 6.5 = 6.5 \wedge 3 \approx 274 \text{ cubic inches}
\end{equation*}
which is still too big. After a few more guesses she tries \(n = 6.2\) so the box would hold
\begin{equation*}
6.2 \times 6.2 \times 6.2 = 6.2 \wedge 3 \approx 238 \text{ cubic inches}
\end{equation*}
Aha! She will make a box that’s approximately \(6.2 \times 6.2 \times 6.2\text{.}\)
That was a lot of guessing. Turns out there’s a name for the answer. We write
\begin{equation*}
n = \sqrt[3]{231}
\end{equation*}
and say that the dimension we are looking for is the 3rd root of 231.
We can use our calculator to find roots. For example, if you have a key that says \(\sqrt[x]{y}\) then you can enter
\begin{equation*}
3 \sqrt[x]{y} ~231 = 6.1357\ldots
\end{equation*}
Different calculators label the roots key differently. For example, you may have to use one of the 2nd, Shift, or Inv key with the \(\wedge\text{,}\) \(y^x\text{,}\) or \(x^y\) key. On a graphing calculator, you may have to enter MATH mode.
For practice, try evaluating the following roots on your calculator. Feel free to ask a classmate or your instructor for help if you can’t figure it out.
\begin{equation*}
\sqrt[3]{125} = 3 \sqrt[x]{y}~ 125= 5
\end{equation*}
\begin{equation*}
\sqrt[3]{1000} = 3 \sqrt[x]{y}~ 1000 = 10
\end{equation*}
\begin{equation*}
\sqrt[4]{81} = 4 \sqrt[x]{y}~ 81 = 3
\end{equation*}
\begin{equation*}
\sqrt[10]{1024} = 10 \sqrt[x]{y}~ {1024} = 2
\end{equation*}
By the way, \(n \wedge 3\) is sometimes called \(n\) cubed and \(\sqrt[3]{~}\) is referred to as the cube root. That terminology comes from the fact that the \(n \times n \times n\) cube has volume \(n \times n \times n = n \wedge 3\text{,}\) as in our example. Also note that the units on volume which are inches \(\times\) inches \(\times\) inches are called cubic inches.
Similarly, \(n \wedge 2\) is called \(n\) squared and \(\sqrt{~}\) (which is shorthand for \(\sqrt[2]{~}\)) is referred to as the square root. Some calculators have a separate key for the square root. That terminology comes from the fact that an \(n \times n\) square has area \(n \times n = n \wedge 2\text{.}\) And a square that was \(n \times n\) in inches would have area measured in inches \(\times\) inches, which are called square inches.
