Skills Refresher:  Exponents and Scientific Notation (6.1 – 6.3) — Intermediate Algebra

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8 Skills Refresher: Exponents and Scientific Notation (6.1 – 6.3) — Intermediate Algebra

6.1 Working With Exponents

Exponents often can be simplified using a few basic properties, since exponents represent repeated multiplication. The basic structure of writing an exponent looks like $x^y,$ where $x$ is defined as the base and $y$ is termed its exponent. For this instance, $y$ represents the number of times that the variable $x$ is multiplied by itself.

When looking at numbers to various powers, the following table gives the numeric value of several numbers to various powers.

$\begin{array}{llllll} \text{Squares}&\text{Cubes}&4^{\text{th}}\text{ Power}&5^{\text{th}}\text{ Power}&6^{\text{th}}\text{ Power}&7^{\text{th}}\text{ Power} \\ \\ 2^2=4&2^3=8&2^4=16&2^5=32&2^6=64&2^7=128 \\ 3^2=9&3^3=27&3^4=81&3^5=243&3^6=729&3^7=2,187 \\ 4^2=16&4^3=64&4^4=256&4^5=1,024&4^6=4,096&4^7=16,384 \\ 5^2=25&5^3=125&5^4=625&5^5=3,125&5^6=15,625&5^7=78,125 \\ 6^2=36&6^3=216&6^4=1,296&6^5=7,776&6^6=46,656&6^7=279,936 \\ 7^2=49&7^3=343&7^4=2,401&7^5=16,807&7^6=117,649&7^7=823,543 \\ 8^2=64&8^3=512&8^4=4,096&8^5=32,768&8^6=262,144&8^7=2,097,152 \\ 9^2=81&9^3=729&9^4=6,561&9^5=59,049&9^6=531,441&9^7=4,782,969 \\ 10^2=100&10^3=1,000&10^4=10,000&10^5=100,000&10^6=1,000,000&10^7=10,000,000 \\ \\ 11^2=121&12^2=144&13^2=169&14^2=196&15^2=225&20^2=400 \end{array}$

For this chart, the expanded forms of the base 2 for multiple exponents is shown:

$\begin{array}{lllllllllllllll} 2^2&=&2&\times &2&=&4,&&&&&&&& \\ 2^3&=&2&\times &2&\times &2&=&8,&&&&&& \\ 2^4&=&2&\times &2&\times &2&\times &2&=&16,&&&& \\ 2^5&=&2&\times &2&\times &2&\times &2&\times &2&=&32&& \\ 2^6&=&2&\times &2&\times &2&\times &2&\times &2&\times &2&=&64\hspace{0.25in} \text{and so on} \\ \end{array}$

Once there is an exponent as a base that is multiplied or divided by itself to the number represented by the exponent, it becomes straightforward to identify a number of rules and properties that can be defined.

The following examples outline a number of these rules.

Example 6.1.1

What is the value of $a^2 \times a^3$ ?

$a^2 \times a^3$ means that you have $(a \times a) (a \times a \times a),$

which is the same as $(a \times a \times a \times a \times a)$

or $a^5$

This means that, when there is the same base and exponent that is multiplied by the same base with a different exponent, the total exponent value can be found by adding up the exponents.

$\text{Product Rule of Exponents: }x^m \times x^n = x^{m+n}$

Example 6.1.2

What is the value of $(a^2)^3$ ?

$(a^2)^3$ means that you have $(a^2) \times (a^2) \times (a^2)$ ,

which is the same as $(a \times a) (a \times a) (a \times a)$

or $(a \times a \times a \times a \times a \times a)$ ,

which equals $a^6$

When you have some base and exponent where both are multiplied by another exponent, the total exponent value can be found by multiplying the two different exponents together.

$\text{Power of a Power Rule of Exponents: }(x^m)^n = x^{mn}$

Example 6.1.3

What is the value of $(ab)^2$ ?

$(ab)^2$ means that you have $(ab) \times (ab)$ ,

which is the same as $(a \times b) \times (a \times b)$

or $(a \times a \times b \times b)$ ,

which equals $a^2b^2$

$\text{Power of a Product Rule of Exponents: }(xy)^n = x^ny^n$

Example 6.1.4

What is the value of $\dfrac{a^5}{a^3}$ ?

$\dfrac{a^5}{a^3}$ means that you have $\dfrac{a \times a \times a \times a \times a}{a \times a \times a}$ , or that you are multiplying $a$ by itself five times and dividing it by itself three times.

Multiplying and dividing by the exact same number is a redundant exercise; multiples can be cancelled out prior to doing any multiplying and/or dividing. The easiest way to do this type of a problem is to subtract the exponents, where the exponents in the denominator are being subtracted from the exponents in the numerator. This has the same effect as cancelling any excess or redundant exponents.

For this example, the subtraction looks like $a^{5-3},$ leaving $a^2.$

$\text{Quotient Rule of Exponents: }\dfrac{x^m}{x^n}=x^{m-n}\hspace{0.25in} (x \ne 0)$

Example 6.1.5

What is the value of $\left(\dfrac{a}{b}\right)^3$ ?

Expanded, this exponent is the same as:

$\dfrac{a}{b}\times \dfrac{a}{b}\times \dfrac{a}{b}$

Which is the same as:

$\dfrac{a \times a \times a}{b \times b \times b} \text{ or } \dfrac{a^3}{b^3}$

One can see that this result is very similar to the power of a product rule of exponents.

$\text{Power of a Quotient Rule of Exponents: }\left(\dfrac{x}{y}\right)^n = \dfrac{x^n}{y^n}\hspace{0.25in} (y \ne 0)$

6.2 Negative Exponents

Consider the following chart that shows the expansion of $a$ for several exponents:

$\begin{array}{lllllllll} a^4\phantom{-}&=&a&\times &a&\times &a&\times &a \\ a^3&=&a&\times &a&\times &a&& \\ a^2&=&a&\times &a&&&& \\ a^1&=&a&&&&&& \\ a^0&=&1&&&&&& \end{array}$

$\begin{array}{lll} a^{-1}&=&\dfrac{1}{a} \\ \\ a^{-2}&=&\dfrac{1}{(a\times a)} \\ \\ a^{-3}&=&\dfrac{1}{(a\times a\times a)} \\ \\ a^{-4}&=&\dfrac{1}{(a\times a\times a\times a)} \end{array}$

If zero and negative exponents are expanded to base 2, the result is the following:

$\begin{array}{llcll} 2^0&=&1&& \\ \\ 2^{-1}&=&\dfrac{1}{2}&& \\ \\ 2^{-2}&=&\dfrac{1}{2\times 2}&\text{or}&\dfrac{1}{4} \\ \\ 2^{-3}&=&\dfrac{1}{2\times 2\times 2}&\text{or}&\dfrac{1}{8} \\ \\ 2^{-4}&=&\dfrac{1}{2\times 2\times 2\times 2}&\text{or}&\dfrac{1}{16} \end{array}$

The most unusual of these is the exponent 0. Any base that is not equal to zero to the zeroth exponent is always 1. The simplest explanation of this is by example.

Example 6.2.1

Simplify $\dfrac{x^3}{x^3}$ .

Using the quotient rule of exponents, we know that this simplifies to $x^{3-3}$ , which equals $x^0.$ And we know

$\dfrac{2^3}{2^3} = \dfrac{8}{8}=1$ ,

$\dfrac{3^3}{3^3} = \dfrac{27}{27}=1$ ,

$\dfrac{4^3}{4^3}=\dfrac{64}{64}=1$ ,

$\dfrac{5^3}{5^3}=\dfrac{125}{125}=1$ ,

and so on. A base raised to an exponent divided by that same base raised to that same exponent will always equal 1 unless the base is 0. This leads us to the zero power rule of exponents:

$\text{Zero Power Rule of Exponents: }x^0 = 1\hspace{0.25in} (x \ne 0)$

This zero rule of exponents can make difficult problems elementary simply because whatever the 0 exponent is attached to reduces to 1. Consider the following examples:

Example 6.2.2

Simplify the following expressions.

$5x^0y^2$
Since $x^0 = 1,$ this simplifies to $5y^2.$
$(5x^0y^2)^0$
Since the zero exponent is on the outside of the parentheses, everything contained inside the parentheses is cancelled out to 1.
$[(15x^3y^2)(25x^2y^2)]^0$
Since the zero exponent is on the outside of the brackets, everything contained inside the brackets cancels out to 1.

When encountering these types of problems, always remain aware of what the zero power is attached to, since only what it is attached to cancels to 1.

When dealing with negative exponents, the simplest solution is to reciprocate the power. For instance:

Example 6.2.3

Simplify the following expressions.

$3x^{-2}y^2$
Since the only negative exponent is $x^{-2}$ , this simplifies to $\dfrac{3y^2}{x^2}.$
$4x^2y^{-3}$
Since the only negative exponent is $y^{-3}$ , this simplifies to $\dfrac{4x^2}{y^3}.$
$(4x^2y^{-3})^{-1}$
Using the power of a power rule of exponents, we get $4^{-1}x^{-2}y^3$ .
Simplifying the negative exponents of $4^{-1}x^{-2}$ , we get $\dfrac{y^3}{4x^2}.$
$(2m^{-1}n^{-3})(2m^{-1}n^{-3})^4$
First using the power of a power rule on $(2m^{-1}n^{-3})^4$ yields $2^4m^{-4}n^{-12}$ .
Now we multiply $2m^{-1}n^{-3}$ by $2^4m^{-4}n^{-12}$ , yielding $2^5m^{-5}n^{-15}$ .
We can write this without any negative exponents as $\dfrac{2^5}{m^5n^{15}}.$

Four Rules of Negative Exponents

$x^{-n}=\dfrac{1}{x^n}\hspace{0.25in} (x \ne 0)$

$\dfrac{1}{x^{-n}}=x^n\hspace{0.25in} (x \ne 0)$

$\left(\dfrac{x}{y}\right)^{-n}=\left(\dfrac{y}{x}\right)^n \hspace{0.25in} (x,y \ne 0)$

$\left(\dfrac{x^ay^b}{z^c}\right)^{-n}=\dfrac{z^{cn}}{x^{an}y^{bn}}\hspace{0.25in} (x,y,z \ne 0)$

6.3 Scientific Notation

Scientific notation is a convenient notation system used to represent large and small numbers. Examples of these are the mass of the sun or the mass of an electron in kilograms. Simplifying basic operations such as multiplication and division with these numbers requires using exponential properties.

Scientific notation has two parts: a number between one and nine and a power of ten, by which that number is multiplied.

$\text{Scientific notation: }a \times 10^b, \text{ where }1 \le a \le 9$

The exponent tells how many times to multiply by 10. Each multiple of 10 shifts the decimal point one place. To decide which direction to move the decimal (left or right), recall that positive exponents means there is big number (larger than ten) and negative exponents means there is a small number (less than one).

Example 6.3.1

Convert 14,200 to scientific notation.

$\begin{array}{rl} 1.42&\text{Put a decimal after the first nonzero number} \\ \times 10^4 & \text{The exponent is how many times the decimal moved} \\ 1.42 \times 10^4& \text{Combine to yield the solution} \end{array}$

Example 6.3.2

Convert 0.0028 to scientific notation.

$\begin{array}{rl} 2.8&\text{Put a decimal after the first nonzero number} \\ \times 10^{-3}&\text{The exponent is how many times the decimal moved} \\ 2.8\times 10^{-3}&\text{Combine to yield the solution} \end{array}$

Example 6.3.3

Convert 3.21 × 10⁵ to standard notation.

Starting with 3.21, Shift the decimal 5 places to the right, or multiply 3.21 by 10⁵.

321,000 is the solution.

Example 6.3.4

Convert 7.4 × 10⁻³ to standard notation

Shift the decimal 3 places to the left, or divide 6.4 by 10³.

0.0074 is the solution.

Working with scientific notation is easier than working with other exponential notation, since the base of the exponent is always 10. This means that the exponents can be treated separately from any other numbers. For instance:

Example 6.3.5

Multiply (2.1 × 10⁻⁷)(3.7 × 10⁵).

First, multiply the numbers 2.1 and 3.7, which equals 7.77.

Second, use the product rule of exponents to simplify the expression 10⁻⁷ × 10⁵, which yields 10⁻².

Combine these terms to yield the solution 7.77 × 10⁻².

Example 6.3.6

(4.96 × 10⁴) ÷ (3.1 × 10⁻³)

First, divide: 4.96 ÷ 3.1 = 1.6

Second, subtract the exponents (it is a division): 10^{4− −3} = 10^{4 + 3} = 10⁷

Combine these to yield the solution 1.6 × 10⁷.

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6.1 Working With Exponents

6.2 Negative Exponents

Four Rules of Negative Exponents

6.3 Scientific Notation

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