# 1.2 Exponents and scientific notation

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In this section students will:
• Use the product rule of exponents.
• Use the quotient rule of exponents.
• Use the power rule of exponents.
• Use the zero exponent rule of exponents.
• Use the negative rule of exponents.
• Find the power of a product and a quotient.
• Simplify exponential expressions.
• Use scientific notation.

Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.

Using a calculator, we enter $\text{\hspace{0.17em}}2,048\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1,536\text{\hspace{0.17em}}×\text{\hspace{0.17em}}48\text{\hspace{0.17em}}×\text{\hspace{0.17em}}24\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3,600\text{\hspace{0.17em}}$ and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately $\text{\hspace{0.17em}}1.3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{13}\text{\hspace{0.17em}}$ bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

## Using the product rule of exponents

Consider the product $\text{\hspace{0.17em}}{x}^{3}\cdot {x}^{4}.\text{\hspace{0.17em}}$ Both terms have the same base, x , but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

The result is that $\text{\hspace{0.17em}}{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}.$

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

${a}^{m}\cdot {a}^{n}={a}^{m+n}$

Now consider an example with real numbers.

${2}^{3}\cdot {2}^{4}={2}^{3+4}={2}^{7}$

We can always check that this is true by simplifying each exponential expression. We find that $\text{\hspace{0.17em}}{2}^{3}\text{\hspace{0.17em}}$ is 8, $\text{\hspace{0.17em}}{2}^{4}\text{\hspace{0.17em}}$ is 16, and $\text{\hspace{0.17em}}{2}^{7}\text{\hspace{0.17em}}$ is 128. The product $\text{\hspace{0.17em}}8\cdot 16\text{\hspace{0.17em}}$ equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

## The product rule of exponents

For any real number $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and natural numbers $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n,$ the product rule of exponents states that

${a}^{m}\cdot {a}^{n}={a}^{m+n}$

## Using the product rule

Write each of the following products with a single base. Do not simplify further.

1. ${t}^{5}\cdot {t}^{3}$
2. ${\left(-3\right)}^{5}\cdot \left(-3\right)$
3. ${x}^{2}\cdot {x}^{5}\cdot {x}^{3}$

Use the product rule to simplify each expression.

1. ${t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}$
2. ${\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}$
3. ${x}^{2}\cdot {x}^{5}\cdot {x}^{3}$

At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.

${x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}$

Notice we get the same result by adding the three exponents in one step.

${x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}$

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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