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Multiplication and Division of Radicals

Rationalize the denominator

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Real World Applications – Algebra I


The United States’ Pentagon – How long is each wall?

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The United States’ Pentagon is the world’s largest office building. It houses the country’s governmental offices and military forces are based. The building itself is a regular pentagon, meaning that all five sides are of equal length. If each outer wall of the Pentagon is \begin{align*}32\sqrt{828} \ feet\end{align*} long, we can apply our knowledge of the addition of radicals to find the total perimeter of the building.

Since we know that every wall is the same length and that there are five walls total, we add \begin{align*}32\sqrt{828}\end{align*} five times. \begin{align*}32 \sqrt{828}+32\sqrt{828}+32\sqrt{828}+32\sqrt{828}+32\sqrt{828}\end{align*}

Since all of the numbers under the radical sign are the same, we treat it the same as adding like terms. We add the “32”, and keep the radicals the same.

\begin{align*}160 \sqrt{828}\end{align*}

Since we’re adding the same side to itself five times, we can also multiply one wall by 5 to get the same answer, instead of adding it to itself five times. We would have \begin{align*}5 \times 32 \sqrt{828}=(5 \times 32) \sqrt{828}=160\sqrt{828}\end{align*}.

We also know that by now, we should be able to simplify radicals. \begin{align*}\sqrt{828}\end{align*} can be simplified. 828’s greatest common factor that’s a perfect square is 36. Using a factor tree or any other method, we have

\begin{align*}32 \sqrt{36 \times 23}=32(6) \sqrt{23}=192\sqrt{23}\end{align*}

Let’s say that for the sake of this exercise we wanted the height of the Pentagon to match the horizontal length of each wall. So, each face of the Pentagon building would be a square. We could calculate this by multiplying \begin{align*}32 \sqrt{828}\end{align*} to itself, or \begin{align*}32 \sqrt{828} \times 32\sqrt{828}\end{align*}. To find the area of each face of the Pentagon, we would multiply everything together. Multiply \begin{align*}32 \times 32\end{align*}, and multiply the numbers in the radical signs. Since we’re multiplying radicals, we can either write \begin{align*}\sqrt{828 \times 828}\end{align*} or \begin{align*}\left(\sqrt{828}\right)^2\end{align*}. Either way, when you multiply a radical by itself, the result is the inside of the root sign. In this case, \begin{align*}\sqrt{828 \times 828} = \left(\sqrt{828}\right)^2 = 828\end{align*}. So, the face of the Pentagon wall would be \begin{align*}32 \times 32 \times 828 = 847,872 \ square \ feet\end{align*}.

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