# 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.

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

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
Hiiii
Chinni
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Ranjay
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ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function