Study Guide: Exponents
What are Exponents?
Exponents are a shorthand way to show how many times a number (called the base) is multiplied to itself.
A number with an exponent is said to be “raised to the power” of that exponent. In the expression, $8^2$, the base of $8$ is raised to the power of the exponent, $2$.
\[ 8^2 = 8 \times 8 = 64 \]
Why use Exponents?
Exponents make it easier to write and calculate multiple-multiplication.
Example:
\[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \]
The Karat Symbol
Sometimes we use the karat symbol ^ (Shift-6 on a computer keyboard) to show an exponent. For example: 2^4 is the same as $2^4$.
\[ \text{2^4 } = 2^4 = 2 × 2 × 2 × 2 = 16 \]
Negative Exponents and Division
Remember that the negative symbol actually means “the opposite.” What is the opposite of “multiplication”?
Keep in mind that although we usually don’t write it, every number can be written as a fraction over 1. For example, 8 means the same thing as $\dfrac{8}{1}$. This makes sense because when we say “eight”, we actually mean “eight ones” or “a group of eight ones”. If you doubt this, test it by putting any other number in the denominator and see if the fraction still equals one-eighth when divided.
When we multiply, we typically work with the numbers above the division line. For example, $8 \times 2 = 16$ means the same thing as:
\[ \dfrac{8}{1} \times \dfrac{2}{1} = \dfrac{8 × 2}{1} = \dfrac{16}{1} \]
Example 1
1 divided by 8.
\[ 8^{-1} = \dfrac{1}{8} = 0.125 \]
Example 2
\[ 5^{-3} = \left(\dfrac{1}{5}\right)^3 = \left(\dfrac{1}{5} \times \dfrac{1}{5} \times \dfrac{1}{5}\right) = \dfrac{1}{5 \times 5 \times 5} = \dfrac{1}{25} = 0.008 \]
Exponents of One
Any number multiplied to the power of 1 equals itself. If the exponent is 1, then the number is not multipled at all, and the value of the expression remains the same.
\begin{align} 1^1 &= 1 \\ 2^1 &= 2 \\ 3^1 &= 3 \\ 9^1 &= 9 \\ 144^1 &= 144 \\ n^1 &= n \end{align}
Exponents of Zero
If any number raised to the first power equals the original number ($n^1 = n$), what is the value of any number (except 0) raised to the zero power? The counter-intuitive answer is 1! For example:
\begin{align} 1^0 &= 1 \\ 2^0 &= 1 \\ 3^0 &= 1 \\ 9^0 &= 1 \\ 144^0 &= 1 \\ n^0 &= 1 \end{align}
Exponent Patterns
This pattern shows why exponents of zero must equal 1. Notice how each expression changes by a multiple of 5. If exponents of zero equal 0, then there would be no way to multiply $0^0 \times 5$ to get 5, and the whole pattern of exponents would break down before it even got started.
\begin{align} &5^3&=&5\times5\times5&=&125 \\ &5^2&=&5\times5&=&25 \\ &5^1&=&5&=&5 \\ &5^0&=&1&=&1 \\ &5^{-1}&=&\dfrac{1}{5}&=&0.2 \\ &5^{-2}&= &\dfrac{1}{5}\times\dfrac{1}{5}&=&0.04 \\ \end{align}
Zero to the Power of Zero?
Because the value of such expressions could be 1 or 0, we say the expression is “indeterminate”.
\[ (0^0 = 1) \text{ and } (0^0 = 0) \]
Therefore:
\[ 0^0 = \text{Indeterminate} \]
Fractional Exponents
Rules
- Roots and exponents are opposites, and undo each other.
- The denominator of a fractional exponent is the root.
- The numerator of a fractional exponent is the exponent of the whole root.
Steps
- Place a radical sign around the base, and move the denominator of the exponent to the index.
- Simplify the root.
- Simplify the exponent.
Example 1
\begin{align} \hline \Large{x^{\frac{{\color{red}n}}{{\color{teal}m}}}} &\Large{ = } \Large{\sqrt[{\color{teal}m}]{x^{\color{red}n}}} \\ \hline \\ \Large{36^{\frac{{\color{red}1}}{{\color{teal}2}}}} &\Large{ = } \Large{\sqrt[{\color{teal}2}]{36^{\color{red}1}}} &\textit{1. Move denominator to index of root} \\[2ex] &\Large{ = } \Large{\sqrt[{\color{teal}2}]{(6)(6)}} &\textit{2. Find the root} \\[2ex] &\Large{ = } \Large{\sqrt[{\color{teal}2}]{6^{\color{teal}2}}} \\[2ex] &\Large{ = } \Large{6} &\textit{3. Simplify exponent} \\[2ex] &\Large{ = } \Large{6} \end{align}
Example 2
\begin{align} \Large{81^{\frac{{\color{red}3}}{{\color{teal}4}}}} &= \Large{\sqrt[{\color{teal}4}]{81}^{{\color{red},3}} } &\textit{1. Move denominator to index of root} \\[2ex] &= \Large{\sqrt[{\color{teal}4}]{{\color{teal}(3)(3)(3)(3)}}^{{\color{red},3}}} &\textit{2. Find the root} \\[2ex] &= \Large{3^{{\color{red}3}} } &\textit{3. Simplify exponent} \\[2ex] &= \Large{27} \end{align}
Putting It All Together
Products with same base
\begin{align} \hline {\color{maroon}a}^n \times {\color{maroon}a}^m &= {\color{maroon}a}^{n+m} \\ \hline \\ {\color{maroon}2}^3 \times {\color{maroon}2}^4 &= {\color{maroon}2}^{3+4} \\ &= {\color{maroon}2}^7 \\ &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 128 \end{align}
Products with same exponent
\begin{align} \hline a^{\color{darkorange}n} \times b^{\color{darkorange}n} &= (a \times b)^{\color{darkorange}n} \\ \hline \\ 3^{\color{darkorange}2} \times 4^{\color{darkorange}2} &= (3 \times 4)^{\color{darkorange}2} \\ &= 12^{\color{darkorange}2} \\ &= 12 \times 12 \\ &= 144 \end{align}
Quotients with same base
\begin{align} \hline \dfrac{{\color{green}a}^n}{{\color{green}a}^m} &= {\color{green}a}^{n – m} \\ \hline \\ \dfrac{{\color{green}2}^5}{{\color{green}2}^3} &= {\color{green}2}^{5–3} \\ &= {\color{green}2}^2 \\ &= 2 \times 2 \\ &= 4 \end{align}
Quotients with same exponent
\begin{align} \hline \dfrac{a^{\color{red}n}}{b^{\color{red}n}} &= \left ( \dfrac{a}{b} \right )^{\color{red}n} \\ \hline \\ \dfrac{4^{\color{red}3}}{2^{\color{red}3}} &= \left ( \dfrac{4}{2} \right )^{\color{red}3} \\ &= 2^{\color{red}3} \\ &= 2 \times 2 \times 2 \\ &= 8 \end{align}
Powers of exponents
\begin{align} \hline (a^{\color{green}n})^{\color{green}m} &= a^{{\color{green}n} \times {\color{green}m}} \\ \hline \\ (2^{\color{green}3})^{\color{green}2} &= 2^{{\color{green}3} \times {\color{green}2}} \\ &= 2^{\color{green}6} \\ &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 64 \end{align}
Powers of exponents II
\begin{align} \hline a^{{\color{red}n^m}} &= a^{({\color{red}n^m})}\\ \hline \\ 2^{{\color{red}3^2}} &= 2^{({\color{ red}3^2})} \\ &= 2^{({\color{red}3\times3})} \\ &= 2^{\color{red}9} \\ &= 2 \times 2 \times 2\times2\times2\times2\times2\times2\times2 \\ &= 512 \end{align}
Negative exponents
\begin{align} \hline b^{{\color{teal}–n}} &= \dfrac{1}{b^{\color{teal}n}} \\ \hline \\ 2^{{\color{teal}–3}} &= \dfrac{1}{2^{\color{teal}3}} \\ &= \dfrac{1}{(2 \times 2 \times 2)} \\ &= \dfrac{1}{8} \\ &= 0.125 \end{align}
Fractional exponents
\begin{align} \hline \Large{x^{\frac{{\color{red}n}}{{\color{teal}m}}}} &\Large{ = } \Large{\sqrt[{\color{teal}m}]{x^{\color{red}n}} } \\ \hline \\ \Large{36^{\frac{{\color{red}1}}{{\color{teal}2}}}} &\Large{ = } \Large{\sqrt[{\color{teal}2}]{36^{\color{red}1}}} \\ \\ &\Large{ = } \Large{36} \end{align}