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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 2,048 × 1,536 × 48 × 24 × 3,600 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 1.3 × 10 13 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 x 3 x 4 . Both terms have the same base, x , but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

x 3 x 4 = x x x 3  factors x x x x 4  factors = x x x x x x x 7  factors = x 7

The result is that x 3 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 a n = a m + n

Now consider an example with real numbers.

2 3 2 4 = 2 3 + 4 = 2 7

We can always check that this is true by simplifying each exponential expression. We find that 2 3 is 8, 2 4 is 16, and 2 7 is 128. The product 8 16 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 a and natural numbers m and n , the product rule of exponents states that

a m 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 t 3
  2. ( −3 ) 5 ( −3 )
  3. x 2 x 5 x 3

Use the product rule to simplify each expression.

  1. t 5 t 3 = t 5 + 3 = t 8
  2. ( −3 ) 5 ( −3 ) = ( −3 ) 5 ( −3 ) 1 = ( −3 ) 5 + 1 = ( −3 ) 6
  3. x 2 x 5 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 x 5 x 3 = ( x 2 x 5 ) x 3 = ( x 2 + 5 ) x 3 = x 7 x 3 = x 7 + 3 = x 10

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

x 2 x 5 x 3 = x 2 + 5 + 3 = x 10
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Questions & Answers

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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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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
Propessor Reply
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
sita Reply
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Propessor
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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
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Propessor
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jai
What is algebra
Pearl Reply
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
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
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Jeffrey
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Jeffrey
I'm going to 11grade
Miranda
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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
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
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Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
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 .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
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Nokwanda
C'est comment
Abdel
Hi
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SORIE
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Chinni
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ANSHU
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Chinni
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Abdel
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
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
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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