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 exponent, $2$, tells us to multiply $8$ twice.
\[ 8^2 = 8 \times 8 = 64 \]
Why use Exponents?
Exponents make it easier to write and calculate multiple-multiplication.
Example:
\[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \]
Watch Out!
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
Remember that the negative symbol actually means “the opposite.”
What is the opposite of “multiplication”?
Example 1
\[ 8^{-1} = \dfrac{1}{8} = 0.125 \]
Example 2
\[ 5^{-3} = \left(\dfrac{1}{5}\right)^3 = \times \dfrac{1}{5} \times \dfrac{1}{5} = 0.008 \]
Example 3
\[ 5^{-3} = \dfrac{1}{5 \times 5 \times 5} = \dfrac{1}{5^3} = \dfrac{1}{125} = 0.008 \]
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 \times 5 \times 5 = 125 \\\ 5^2 &= 5 \times 5 = 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}