Exponential Expressions are ones where the base is powered by an exponent. An exponent indicates by how many times the base multiplies itself. An exponential expression is commonly represented as:
where \(a\) represents the base and \(m\) represents the exponent.
For instance, we can simplify the expresssion \(4^2\) as such:
This example represents the base (4) being raised to an exponent (2) and multiplying itself 2 times.
Below is a list of all the exponent rules that will come in handy when encountering and simplifying different types of exponential expressions:
| Name | Law | Example |
|---|---|---|
| Product | \(\textcolor{red}{a}^\textcolor{green}{m}\cdot\textcolor{red}{a}^\textcolor{green}{n}=\textcolor{red}{a}^{\textcolor{green}{m+n}}\) | \(\textcolor{red}{2}^\textcolor{green}{3}\cdot\textcolor{red}{2}^\textcolor{green}{5}=\textcolor{red}{2}^{\textcolor{green}{3+5}}=\textcolor{red}{2}^{\textcolor{green}{8}}=256\) |
| Quotient | \(\cfrac{\textcolor{red}{a}^\textcolor{green}{m}}{\textcolor{red}{a}^\textcolor{green}{n}}=\textcolor{red}{a}^{\textcolor{green}{m-n}}\) | \(\cfrac{\textcolor{red}{3}^\textcolor{green}{7}}{\textcolor{red}{3}^\textcolor{green}{4}}=\textcolor{red}{3}^{\textcolor{green}{7-4}}=\textcolor{red}{3}^\textcolor{green}{3}=27\) |
| Power of a Power | \((\textcolor{red}{a}^\textcolor{green}{m})^\textcolor{green}{n}=\textcolor{red}{a}^{\textcolor{green}{mn}}\) | \((\textcolor{red}{2}^\textcolor{green}{5})^\textcolor{green}{2}=\textcolor{red}{2}^{(\textcolor{green}{5})(\textcolor{green}{2})}=\textcolor{red}{2}^{\textcolor{green}{10}}=1024\) |
| Power of a Product | \((\textcolor{red}{ab})^\textcolor{green}{m}=\textcolor{red}{a}^\textcolor{green}{m}\textcolor{red}{b}^\textcolor{green}{m}\) | \((\textcolor{red}{5x}^\textcolor{green}{3})^\textcolor{green}{5}=\textcolor{red}{5}^\textcolor{green}{5}\textcolor{red}{x}^{(\textcolor{green}{3})(\textcolor{green}{5})}=125x^{15}\) |
| Power of a Quotient | \((\cfrac{\textcolor{red}{a}}{\textcolor{red}{b}})^\textcolor{green}{m}=\cfrac{\textcolor{red}{a}^\textcolor{green}{m}}{\textcolor{red}{b}^\textcolor{green}{m}}\) | \(\left(\cfrac{\textcolor{red}{4x}}{\textcolor{red}{2}}\right)^\textcolor{green}{4}=\cfrac{\textcolor{red}{4}^\textcolor{green}{4}\textcolor{red}{x}^\textcolor{green}{4}}{\textcolor{red}{2}^\textcolor{green}{4}}=\cfrac{\textcolor{red}{256x}^\textcolor{green}{4}}{\textcolor{red}{16}}=16x^4\) |
| Negative Exponent | \(\textcolor{red}{a}^{\textcolor{green}{-m}}\) \(=\cfrac{1}{\textcolor{red}{a}^\textcolor{green}{m}}\) | \(\textcolor{red}{9}^{\textcolor{green}{-2}}=\cfrac{1}{\textcolor{red}{9}^\textcolor{green}{2}}=\cfrac{1}{81}\) |
| Fractional Exponent | \(\textcolor{red}{a}^{\textcolor{green}{m/n}}=\sqrt[\textcolor{green}{n}]{\textcolor{red}{a}^\textcolor{green}{m}}\) | \(\textcolor{red}{5}^{\textcolor{green}{9/3}}=\sqrt[\textcolor{green}{3}]{\textcolor{red}{5}^\textcolor{green}{9}}=125\) |
| Zero Exponent | \(\textcolor{red}{a}^\textcolor{green}{0}=1\) for \(\textcolor{red}{a} ≠ 0\) | \(\textcolor{red}{2}^\textcolor{green}{0}=1\); \(\textcolor{red}{0}^\textcolor{green}{0}=\) ERROR |
| Zero Base | \(\textcolor{red}{0}^\textcolor{green}{m}=0\) for \(0 < \textcolor{green}{m}\) | \(\textcolor{red}{0}^\textcolor{green}{3}=0\); \(\textcolor{red}{0}^{\textcolor{green}{-0.5}}=\) ERROR |
NOTE: When simplifying expressions, it's recommended that you use the Power of a Power Law first and the Negative Exponent Law last.
Simplify the expression \(3x^3\cdot4x^4\).
We can multiply the bases of the both terms. After, we can combine the exponents of both terms:
\(= \textcolor{red}{3}x^\textcolor{green}{3}\cdot\textcolor{red}{4}x^\textcolor{green}{4}\)
\(= \textcolor{red}{12}x^{\textcolor{green}{7}}\)
Therefore, we can determine that \(3x^3\cdot4x^4\) simplified is \(\boldsymbol{12x^7}\).
\(x^2(x^3)=1024\)
First, we can use Product Law (\(a^m\cdot a^n=a^{m+n}\)) in order to add the exponents (\(2\) and \(3\)) of their common base (\(x\)).
\(\textcolor{red}{x}^\textcolor{green}{2}(\textcolor{red}{x}^\textcolor{green}{3})=1024\)
\(\textcolor{red}{x}^{\textcolor{green}{5}}=1024\)
In order to solve for \(x\), we can take the root of both sides:
\(\sqrt[\textcolor{blue}{5}]{\textcolor{red}{x}^{\textcolor{green}{5}}}=\sqrt[\textcolor{blue}{5}]{1024}\)
\(x = 4\)
Therefore, we can determine that \(\boldsymbol{x = 4}\).
\(\cfrac{8x^6}{12x^4}=54\)
We can first multiply both sides by the equations reciprocol in order to cancel out its coefficient:
\(\cfrac{\textcolor{red}{8}x^6}{\textcolor{red}{12}x^4}=\textcolor{red}{54}\)
\(\left(\cfrac{\cancel{\textcolor{red}{8}}x^6}{\cancel{\textcolor{red}{12}}x^4}\right)\left(\cancel{\cfrac{12}{8}}\right)=(\textcolor{red}{54})\left(\cfrac{12}{8}\right)\)
\(\cfrac{\textcolor{red}{1}x^6}{\textcolor{red}{1}x^4}=\textcolor{red}{81}\)
We can further simplify the equation using Quotient Law \(\left(\cfrac{a^m}{a^n}=a^{m-n}\right)\) to find the simplified exponent value:
Finally, we can solve for \(x\) by taking the square root of both sides:
\(\textcolor{green}{x^2}=\textcolor{red}{81}\)
\(\sqrt{\textcolor{green}{x^2}}=\sqrt{\textcolor{red}{81}}\)
\(x = 9\)
Therefore, we can determine that \(\boldsymbol{x = 9}\).
These can be described as exponents that are represented as integers (or whole numbers) which can be either positive or negative.
As stated above, Positive integers (i.e. \(5\), \(8\), \(13\)) indicate by how many times a base should multiply itself. Conversely, Negative Integers (i.e. \(-4\), \(-6\), \(-9\)) specify to first move the base to its opposite side (i.e. if it's on the numerator, move it to the denominator and vice versa). This will cause the exponent to go from negative to positive. Afterwards, multiply the base by itself \(m\) amount of times.
For instance, we can simplify the expression \(3^{-2}\) as such:
This example represents the base (\(3\)) being moved to the denominator and then multiplying itself 2 times.
Additionally, we can simplify the expression \(= \cfrac{2^3}{2^5}\) as such:
This example represents the exponents (\(3\) and \(5\)) being subtracted as the bases are the same (\(2\)). Since the exponent is left as negative, the base is switched to the denominator and then raised to its power.
Simplify \(\cfrac{3x^{-2}}{(2y)^{-1}}\). Leave everything with exact numbers with positive exponent answers.
First, we can use Power of a Product Rule (\((ab)^m=a^mb^m\)) to multiply the entire denominator by \(-1\):
\(= \cfrac{3x^{-2}}{(2^\textcolor{red}{1}y^\textcolor{red}{1})^{\textcolor{red}{-1}}}\)
\(=\cfrac{3x^{-2}}{2^{\textcolor{red}{-1}}y^{\textcolor{red}{-1}}}\)
We can then shift all values with negative exponents to the opposite side:
\(=\cfrac{3\textcolor{red}{x^{-2}}}{\textcolor{green}{2^{-1}}\textcolor{blue}{y^{-1}}}\)
\(= \cfrac{(3)(\textcolor{green}{2^1})(\textcolor{blue}{y^1})}{\textcolor{red}{x^2}}\)
Finally, we can simplify the expression by multiplying the coefficients on the numerator:
\(= \cfrac{(\textcolor{red}{3})(\textcolor{red}{2})(y)}{x^2}\)
\(= \cfrac{\textcolor{red}{6}y}{x^2}\)
Therefore, we can determine that \(\cfrac{3x^{-2}}{(2y)^{-1}}\) simplified is \(\boldsymbol{\cfrac{6y}{x^2}}\).
\((3d^{-3})^3\cdot3d^{-2}\)
First, we can use Power of a Product Law \((ab)^m=a^mb^m\) to multiply the first term, \(3d^{-3}\), by its exponent, \(3\):
\((3^{\textcolor{red}{1}}d^{\textcolor{red}{-3}})^{\textcolor{red}{3}}\cdot3d^{-2}\)
\(= 3^{\textcolor{red}{3}}d^{\textcolor{red}{-9}} \cdot 3d^{-2}\)
Then, we can use Product Law (\(a^m.a^n=a^{m+n}\)) to add the exponents of like bases (\(3\) and \(d\)):
\(= 3^{\textcolor{red}{3}}d^{\textcolor{blue}{-9}} \cdot 3^{\textcolor{red}{1}}d^{\textcolor{blue}{-2}}\)
\(= 3^{\textcolor{red}{4}}d^{\textcolor{blue}{-11}}\)
We can fully simplify by moving \(d\) to the denominator and raising \(3\) to its exponent:
\(=\textcolor{red}{3^4}\textcolor{green}{d^{-11}}\)
\(= \cfrac{\textcolor{red}{81}}{\textcolor{green}{d^{11}}}\)
Therefore, we can determine that \((3d^{-3})^3 \cdot 3d^{-2}\) simplified is \(\boldsymbol{\cfrac{81}{d^{11}}}\).
\(\left[\cfrac{(-2a^{-2})^3a^3}{4a^{-4}}\right]^{-3}\)
First, we can use Power of a Product Law (\((ab)^m=a^mb^m\)) by multiply the first portion of the numerator (\(-2a^{-2}\)) by its exponent (\(3\)):
\(= \left[\cfrac{(-2a^{\textcolor{red}{-2}})^{\textcolor{red}{3}}a^3}{4a^{-4}}\right]^{-3}\)
\(= \left[\cfrac{(-2)^{\textcolor{red}{3}}a^{\textcolor{red}{-6}}a^3}{4a^{-4}}\right]^{-3}\)
Next, we can use Quotient Law \(\left(\cfrac{a^m}{a^n}=a^{m-n}\right)\) to subtract the exponents for the \(a\) terms on the numerator:
\(= \left[\cfrac{(-2)^{3}a^{\textcolor{red}{-6}}a^\textcolor{red}{3}}{4a^{-4}}\right]^{-3}\)
\(= \left[\cfrac{(-2)^{3}a^{\textcolor{red}{-3}}}{4a^{-4}}\right]^{-3}\)
Then, we can use Power of a Quotient Law \(\left(\left(\cfrac{a}{b}\right)^m=\cfrac{a^m}{b^m}\right)\) to raise the whole expression to the same exponent (\(-3\)):
\(= \left[\cfrac{(-2)^{\textcolor{red}{3}}a^{\textcolor{red}{{-3}}}}{4^{\textcolor{red}{1}}a^{\textcolor{red}{-4}}}\right]^{\textcolor{red}{-3}}\)
\(= \cfrac{(-2)^{\textcolor{red}{-9}}a^{\textcolor{red}{9}}}{4^{\textcolor{red}{-3}}a^{\textcolor{red}{12}}}\)
After, we can shift all values with negative exponents to opposite positions in the expression:
\(= \cfrac{(\textcolor{red}{-2})^{\textcolor{green}{-9}}a^{9}}{\textcolor{red}{4}^{\textcolor{green}{-3}}a^{12}}\)
\(= \cfrac{\textcolor{red}{4}^{\textcolor{green}{3}}a^9}{(\textcolor{red}{-2})^{\textcolor{green}{9}}a^{12}}\)
Finally, we can fully simplify by exponentiating the numerical bases and finding the lowest common denominator. We can also use Quotient Law \(\left(\cfrac{a^m}{a^n}=a^{m-n}\right)\) for subtracting the exponents for the \(a\)-values:
\(= \cfrac{\textcolor{red}{4}^{\textcolor{red}{3}}\textcolor{green}{a^9}}{(\textcolor{red}{-2})^{\textcolor{red}{9}}\textcolor{green}{a^{12}}}\)
\(= \cfrac{\textcolor{red}{64}{\textcolor{green}{a^{-3}}}}{\textcolor{red}{-512}}\)
\(= \cfrac{\textcolor{red}{64}}{\textcolor{red}{-512}{\textcolor{green}{a^3}}}\)
\(= \cfrac{\textcolor{red}{1}}{\textcolor{red}{-8}{\textcolor{green}{a^3}}}\)
Therefore, we can determine that \(\left[\cfrac{(-2a^{-2})^3a^3}{4a^{-4}}\right]^{-3}\) simplified is \(\boldsymbol{\cfrac{1}{-8a^3}}\).