1.2: Exponents and Scientific Notation
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- Various rules of Exponents
- 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
Using a calculator, we enter
Using the Product Rule of Exponents
Consider the product
The result is that
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.
Now consider an example with real numbers.
We can always check that this is true by simplifying each exponential expression. We find that
For any real number a and natural numbers
Write each of the following products with a single base. Do not simplify further.
Solution
Use the product rule (Equation
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.
Notice we get the same result by adding the three exponents in one step.
Write each of the following products with a single base. Do not simplify further.
- Answer a
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- Answer b
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- Answer c
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Using the Quotient Rule of Exponents
The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
For the time being, we must be aware of the condition
For any real number
Write each of the following products with a single base. Do not simplify further.
- Answer a
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- Answer b
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- Answer c
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Using the Power Rule of Exponents
Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression
The exponent of the answer is the product of the exponents:
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.
Product Rule | Power Rule |
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For any real number a and positive integers m and n, the power rule of exponents states that
Write each of the following products with a single base. Do not simplify further.
- Answer a
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- Answer b
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- Answer c
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Using the Zero Exponent Rule of Exponents
Return to the quotient rule. We made the condition that
If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is
The sole exception is the expression
For any nonzero real number a, the zero exponent rule of exponents states that
Simplify each expression using the zero exponent rule of exponents.
Solution
Use the zero exponent and other rules to simplify each expression.
a.
b.
c.
d.
Simplify each expression using the zero exponent rule of exponents.
- Answer a
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- Answer b
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- Answer c
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- Answer d
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Using the Negative Rule of Exponents
Another useful result occurs if we relax the condition that
Divide one exponential expression by another with a larger exponent. Use our example,
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression an is defined when
For any nonzero real number a and natural number n, the negative rule of exponents states that
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
Solution
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
- Answer a
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- Answer b
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- Answer c
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Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
Solution
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
- Answer a
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- Answer b
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Finding the Power of a Product
To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
In other words,
For any real numbers a and b and any integer n, the power of a product rule of exponents states that
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
Solution
Use the product and quotient rules and the new definitions to simplify each expression.
Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.
- Answer a
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- Answer b
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- Answer c
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- Answer d
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- Answer e
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Finding the Power of a Quotient
To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.
For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
Solution
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.
- Answer a
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- Answer b
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- Answer c
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- Answer d
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- Answer e
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Simplifying Exponential Expressions
Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.
Simplify each expression and write the answer with positive exponents only.
Solution
Using Scientific Notation
Recall at the beginning of the section that we found the number
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of
For example, consider the number
We obtain
Working with small numbers is similar. Take, for example, the radius of an electron,
Be careful not to include the leading
A number is written in scientific notation if it is written in the form
Write each number in scientific notation.
- Distance to Andromeda Galaxy from Earth:
- Diameter of Andromeda Galaxy:
- Number of stars in Andromeda Galaxy:
- Diameter of electron:
- Probability of being struck by lightning in any single year:
Solution
a.
b.
c.
d.
e.
Analysis
Observe that, if the given number is greater than
Write each number in scientific notation.
- U.S. national debt per taxpayer (April 2014):
- World population (April 2014):
- World gross national income (April 2014):
- Time for light to travel
- Probability of winning lottery (match
of possible numbers):
- Answer a
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- Answer b
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- Answer c
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- Answer d
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- Answer e
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Converting from Scientific to Standard Notation
To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal n places to the right if
Convert each number in scientific notation to standard notation.
Solution
a.
b.
c.
d.
Convert each number in scientific notation to standard notation.
- Answer a
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- Answer b
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- Answer c
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- Answer d
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Using Scientific Notation in Applications
Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in
When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product
Perform the operations and write the answer in scientific notation.
Solutions
Perform the operations and write the answer in scientific notation.
- Answer a
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- Answer b
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- Answer c
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- Answer d
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- Answer e
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In April 2014, the population of the United States was about
Solution
The population was
The national debt was
To find the amount of debt per citizen, divide the national debt by the number of citizens.
The debt per citizen at the time was about
An average human body contains around
- Answer
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Number of cells:
; length of a cell: ; total length: or .
Access these online resources for additional instruction and practice with exponents and scientific notation.
Converting to Decimal Notation
Key Equations
Product rule | |
Quotient rule | |
Power rule | |
Zero exponent rule | |
Negative rule | |
Power of a product rule | |
Power of a quotient rule |
Key Concepts
- Products of exponential expressions with the same base can be simplified by adding exponents. See Example.
- Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example.
- Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example.
- An expression with exponent zero is defined as 1. See Example.
- An expression with a negative exponent is defined as a reciprocal. See Example and Example.
- The power of a product of factors is the same as the product of the powers of the same factors. See Example.
- The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example.
- The rules for exponential expressions can be combined to simplify more complicated expressions. See Example.
- Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example and Example.
- Scientific notation may be used to simplify calculations with very large or very small numbers. See Example and Example.