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Exponential Properties Involving Products

Add exponents to multiply exponents by other exponents

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Real World Applications – Algebra I


Using exponents to figure out the wingspan of bats

Student Exploration

Exponents can be used in a variety of ways to represent length. Specifically, negative exponents are used to represent how small something is. Bats, for example, are pretty tiny creatures. Zoologists use negative exponents to measure different parts of bats, such as their wingspan. The world’s smallest bat, the bumblebee bat, for example, weighs of \begin{align*}7 \times 10^{-2}\end{align*}7×102 ounces. Why would this measurement be easier to compare than 0.07 ounces?

Let’s compare this weight with the weight of other animals. A dragonfly, has a weight of 0.0001 ounce, or \begin{align*}1 \times 10^{-4}\end{align*}1×104 ounce. Is this heavier, or lighter than the weight of the bumblebee bat?

It’s beneficial to look at the scientific notation of the numbers and compare, rather than looking at the decimals. We know that \begin{align*}10^{-4}\end{align*}104 is smaller than \begin{align*}10^{-2}\end{align*}102, so we know that the dragonfly is lighter. But by how much?

We can find out by figuring out what to multiply \begin{align*}10^{-4}\end{align*}104 by to get \begin{align*}10^{-2}\end{align*}102. If we keep multiplying \begin{align*}10^{-4}\end{align*}104 by 10 two times, we can get to \begin{align*}10^{-2}\end{align*}102. We’re multiplying by \begin{align*}10^2\end{align*}102, or by 100. This means that the weight of the bat is 100 times heavier than the dragonfly. We knew that a bat is heavier than a dragonfly, but in this way we can quantify the weight differences. We know that \begin{align*}10^{-4} \times 10^2 = 10^{-2}\end{align*}104×102=102 because when we multiply two numbers with the same base, we can add the exponents. We can also verify this by dividing \begin{align*}10^{-2}\end{align*}102 by \begin{align*}10^2\end{align*}102. Knowing that when dividing and then bases are the same, we can subtract the exponents. \begin{align*}-2 - 2 = -4\end{align*}22=4, which is the exponent for what we started with.

Extension Investigation

Compare the weight or another part of two animals and represent their measurements in scientific notation. How can you use the laws of exponents to compare these measurements?

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