Section 1.C Exponents and Scientific Notation
¶Objectives
Subsection 1.C.1 Exponent Properties
Recall from Section 0.0 that exponents represent repeated multiplication:
- \(\displaystyle x^1 = x\)
- \(\displaystyle y^2 = (y)(y)=yy\)
- \(\displaystyle 2^3 = (2)(2)(2)=8\)
- \(\displaystyle 10^4 = (10)(10)(10)(10)=10,000\)
There are several rules that simplify operations involving exponents.
Example 1.C.1. Multiplication with Exponents.
Simplify.
WeBWorK: Entering Exponents.
Type a^5 for \(a^5.\)
Notice that the exponents \(3\) and \(2\) give the number of times to multiply \(a,\) and the product is the total number of times \(a\) gets multiplied, \(5:\) \(a^3 a^2=a^{3+2}=a^5\text{.}\) This is known as the product rule of exponents:
The product rule of exponents can be used to multiply powers with the same base. This is shown in the following examples:
Example 1.C.2. Product Rule of Exponents.
Simplify.
Example 1.C.3. Product Rule of Exponents.
Simplify.
When multiplication and addition/subtraction are used in the same expression, you need to keep track of which rule to use.
Example 1.C.4. Combine like terms.
Write the expression without parenthesis. Combine like terms.
Next, we consider what happens when we divide with exponents.
Example 1.C.5. Division with Exponents.
Simplify.
Since the exponent in the denominator reduces the exponent in the numerator, we may simplfy the quotient by subtracting the exponents, \(\dfrac{a^5}{ a^2} = a^{5-2}= a^3\text{.}\) This is known as the quotient rule of exponents.
The quotient rule of exponents may be used to divide powers by subtracting exponents on like bases. This is shown in the following examples:
Example 1.C.6. Quotient Rule of Exponents.
Simplify:
A power may itself be raised to a second exponent.
Example 1.C.7. Powers with Exponents.
Simplify.
Since we have \(a^2\) times itself \(3\) times, we can multiply the exponents \((a^2)^3 = a^{2\cdot 3} = a^6.\) This is known as the power of a power rule of exponents
There are two more useful properties of exponents.
Example 1.C.8. Powers of Products.
Simplify.
We see that the exponent \(3\) applies to each factor in the product: \((ab)^3 = a^3b^3\text{.}\) This is known as the power of a product rule of exponents:
It is important that we only use the power of a product rule with multiplication inside parentheses. This property does NOT work if there is addition or subtraction inside the parentheses.
| WARNING |
| \((a+b)^n\) is NOT equal to \(a^n+b^n\) if \(n\ne 1\text{,}\) beware of this error! |
| For example, \((1+3)^2 = (4)^2 = 16\) but \(1^2 + 3^2 = 1 + 9 = 10.\) |
Another property that is very similar to the power of a product rule is considered next.
Example 1.C.9. Powers of Quotients.
Simplify.
We see that the exponent applies to each factor in both the numerator and denominator, \(\left(\frac{a}{b} \right)^3=\frac{a^3}{b^3}\text{.}\) This is known as the power of a quotient rule of exponents:
We may need to use several rules to simplify an expression. This is shown in the following examples:
Example 1.C.10. Simplify Exponents.
Simplify.
WeBWorK: Entering Multi-digit Exponents.
If your exponent has two or more digits, you must place parentheses around it. Otherwise, WeBWorK only takes the first number after the ^ as the exponent: Type x^(12) for \(x^{12}\text{.}\)
Example 1.C.11. Simplify Exponents.
Simplify.
When we multiply powers its important to remember that we add or subtract exponents, not bases. An expression such as \(5^3\) does not mean we multiply \(5\) by \(3\text{,}\) rather we multiply \(5\) three times, \(5 \cdot 5 \cdot 5 = 125\text{.}\)
Example 1.C.12. Simplify Exponents.
Simplify.
In the previous example we did not multipy \(4\) by the exponent \(3\) to get \(12\text{:}\) this would have been incorrect. Never multiply a base by the exponent.
Generally, simplifying an expression requires a combination of properties, and deciding which property to use first may be confusing. However, the order of operations always applies and can be your guide. You should simplify inside any parentheses first, then simplify any exponents (using power rules), and finally simplify any multiplication or division (using product and quotient rules).
Example 1.C.13. Simplify Exponents.
Simplify.
Example 1.C.14. Simplify Exponents.
Simplify.
Example 1.C.15. Simplify Exponents.
Simplify.
Example 1.C.16. Simplify Exponents.
Simplify.
Remember to follow the order of operations as a guide: simplify inside parentheses first, then use the power rules, then the product and quotient rules.
Checkpoint 1.C.17. Simplify Exponents.
Subsection 1.C.2 Negative Exponents
Next we consider zero and negative exponents.
Example 1.C.18. Investigate Zero Power.
When we combine the two solutions we get:
This final result is an important property known as the Zero Power Rule of Exponents:
Any number or expression raised to the zero power (except \(0\)) will always be \(1\text{.}\) The problem with \(0\text{,}\) is that \(0\) raised to any positive integer \(n\) is zero. For example, \(0^3 = 0 \cdot 0 \cdot 0 = 0\) and so there is disagreement as to what \(0^0\) should be. We will avoid the issue by declaring it to be not defined and not include it in any of the exercises.
Example 1.C.19. Zero Power Rule of Exponents.
Simplify.
Example 1.C.20. Investigate Negative Exponents.
Since both expressions are solutions, they must be equal:
This example illustrates an important property of exponents. Negative exponents indicate the reciprocal of the base. We use our knowledge of how to divide fractions to understand how to simplify the following:
We summarize the rule for simplifying negative exponents:
It is important to note a negative exponent does not mean the expression is negative, only that we need the reciprocal of the base.
Example 1.C.21. Simplify a Fraction with a Negative Exponent.
Simplify.
The exponent properties are summarized in the following table:
| Rules of Exponents | |
| Product Rule of Exponents | \(a^m a^n=a^{m+n}\) (1.C.1) |
| Quotient Rule of Exponents | \(\frac{a^m}{a^n} =a^{m-n}\) (1.C.2) |
| Power of a Power Rule of Exponents | \((a^m)^n=a^{mn}\) (1.C.3) |
| Power of a Product Rule of Exponents | \((ab)^m=a^m b^m\) (1.C.4) |
| Power of a Quotient Rule of Exponents | \(\left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\) (1.C.5) |
| Zero Power Rule of Exponents | \(a^0=1\) (1.C.6) |
| Negative Power Rule of Exponents | \(a^{-n}=\frac{1}{a^n} \text{ and } \frac{1}{a^{-n}} = a^n \) (1.C.7) |
The rules of exponents apply to negative exponents as well as positive exponents. It is convenient to keep the negative exponents until the end of the problem and then "move them around" to their correct location (numerator or denominator). It is important to be very careful of rules for adding, subtracting, and multiplying with negatives.
Example 1.C.22. Negative Exponents.
Simplify.
WeBWorK: Entering Simplified Expressions.
Type -x^3/(2y^(12)) for \(\dfrac{-x^3}{2 y^{12}}\text{.}\)
Note the two sets of parentheses: one set around the \(12\) and the other around the denominator. Without the parentheses, -x^3/2y^12 the expression is interpreted as \(\dfrac{-x^3}{2}y^1 2\text{.}\)
Simplified expressions require a single fraction form: (num)/(denom) with
- only integer coefficients: NOT \(\dfrac{-0.5x^3}{y^{12}}\)
- negative signs placed in numerator: NOT \(\dfrac{x^3}{-2 y^{12}}\)
- no negative exponents: NOT \(\frac{-1}{2}x^3y^{-12}\)
We now have the following nine properties of exponents.
| Properties of Exponents | ||
| \(a^m a^n=a^{m+n}\) | \((ab)^m=a^m b^m\) | \(a^{-m}=\dfrac{1}{a^m}\) |
| \(\frac{a^m}{a^n}=a^{m-n}\) | \(\left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\) | \(\dfrac{1}{a^{-m}}= a^m\) |
| \((a^m)^n=a^{mn}\) | \(a^0=1\) | \(\left( \dfrac{a}{b}\right)^{-m}=\dfrac{b^m}{a^m}\) |
Checkpoint 1.C.23. Simplify Negative Exponents.
Checkpoint 1.C.24. Simplify Negative Exponents with Powers.
Subsection 1.C.3 Scientific Notation
Scientific notation is used to represent really large or really small numbers. The distance that light travels in a year in miles is a really large number. The mass of a single hydrogen atom in grams is a really small number. Doing basic operations such as multiplication and division with these numbers would normally be very cumbersome. However, our exponent properties make this process much simpler.
Scientific notation is written as the product of two numbers:
- A number between one and ten (it can be equal to one, but not ten), and
- Ten to some integer power or exponent.
Recall that multiplying by \(10\) in effect moves the decimal point one place. Here we multiply by \(10\) twice and the decimal moves two places: \(3.2 \times 10^2 = 3.2 \times 100 = 320.\) So the exponent tells how many places the decimal moves when changing between scientific notation and standard notation.
Keeping this in mind, we can easily make conversions between standard notation and scientific notation.
Example 1.C.25. Convert Standard to Scientific: Large number.
When scientific notation is used to represent small numbers, the format requires the number be rewritten as a "bigger" value: \(0.0021\) must be written as \(2.1 \times 10^b\text{.}\) What should the value of the exponent, \(b,\) be? Note that the decimal was moved three places and we've changed the value to be bigger than the original number. Here is where the exponent on the \(10\) comes to the rescue. If we divide \(2.1\) by \(10\) three times, the result will be \(.0021\text{.}\) Dividing by \(10\) is the same thing as multipying by \(\frac{1}{10}\) and negative exponents allow us to write the fraction as a power of \(10\text{:}\) \(10^{-1} = \frac{1}{10}.\) To finish the conversion of \(0.0021\) to scientific notation: \(0.0021 = 2.1 \times 10^{-3}.\)
Example 1.C.26. Convert Standard to Scientific: Small number.
To decide which direction to move the decimal (left or right) we simply need to remember that positive exponents mean we have a big number (bigger than ten) in standard notation, and negative exponents mean we have a small number (less than one) in standard notation. The negative exponent simply informs us that we are dealing with a small number.
WeBWorK: Entering Scientific Notation.
Use lower case x for the times sign \(\times\text{:}\) Enter 1.42 x 10^4 for \(1.42 \times 10^4.\)
For negative exponents or exponents with more than one digit in scientific notation, WeBWorK has a different rule than for general exponents. In both cases do NOT put parentheses around the exponent: Enter 4.2 x 10^-3 for \(4.2 \times 10^{-3}.\)
For exponents with two or more digits, enter 7.24 x 10^21 for \(7.24 \times 10^{21}.\)
IMPORTANT: When entering scientific notation, you must first select the "Tt" option in the math menu (MathQuill) before typing the number.
Checkpoint 1.C.27. Write in Scientific Notation.
Example 1.C.28. Convert Scientific to Standard.
Example 1.C.29. Convert Scientific to Standard.
Checkpoint 1.C.30. Write in Standard/Decimal form.
When you use a calculator or computer, sometimes scientific notation will be displayed a little differently. The number will look like 2.341 E03 or 2.341 e03. In this case, the E03 is short for "\(\times 10^3\)" and the number is \(2.341 \times 10^3\text{.}\)
Example 1.C.31. Interpret Computer/Calculator Notation.
Checkpoint 1.C.32. Interpret Calculator/Computer notation.
Subsection 1.C.4 Multiplication and Division Using Scientific Notation
To simplify a product or quotient of numbers in scientific notation, note that all the numbers are multiplied or divided so the numbers can be combined in any order to get the same answer. In this case, we multiply or divide the first numbers, then use exponent properties to simplify the powers of \(10\text{.}\) Consider the following examples:
Example 1.C.33. Multiplication with Scientific Notation.
Example 1.C.34. Division with Scientific Notation.
Example 1.C.35. Powers with Scientific Notation.
After computation, if the front number is greater than \(10,\) we convert the front number into scientific notation and then simplify the expression.
Example 1.C.36. Multiplication with Scientific Notation.
Checkpoint 1.C.37. Multiply Two Numbers in Scientific Notation.
Example 1.C.38. Division with Scientific Notation.
Checkpoint 1.C.39. Divide Two Numbers in Scientific Notation.
Subsection 1.C.5 Addition and Subtraction Using Scientific Notation
To add or subtract numbers expressed in scientific notation we could first write both numbers in standard notation, perform the operation, and then convert back to scientific notation.
Converting back and forth between scientific notation and decimals is not very efficient especially if the exponents are very large or very small. Because both numbers were times \(10^6\text{,}\) we may combine like terms involving \(10^6\text{,}\) by adding the front numbers, then writing the result in scientific notation.
Example 1.C.40. Addition with Scientific Notation.
Subtraction of numbers written in scientific notation is similar if the exponents on the \(10\)'s are equal.
Example 1.C.41. Subtraction with Scientific Notation.
Subtract and write the result using scientific notation.
This method of adding/subtracting runs into trouble if the scientific numbers do not both involve the same power of \(10\text{.}\) For example, we cannot just add the number parts for the following: \((2.1\times 10^2)+(3.5 \times 10^6)\) since \(2.1\times 10^2=210\) and \(3.5 \times 10^6=3,500,000\text{.}\) The number parts do not match up once we apply the powers of \(10\text{.}\) To do the calculation in scientific notation, we need to adjust one of the numbers to match the power of \(10\) of the other number. Usually, it is easier to match the smaller exponent.