Like other operations, exponents can apply to many different types of numbers. However, certain types of numbers change the properties of the exponential notation in question.
Before calculating the powers of the different exponential notations, it is important to understand the definition of exponential notation and the meaning of its components.
||a^n = \underbrace{a \times a \times a \times ... \times a}_{n \ \text{times}}||
The value of the exponent represents the number of times the base is multiplied by itself.
||\begin{align} 6^4 &= \underbrace{6 \times 6 \times 6 \times 6}_{4 \ \text{times}} \\ &=1296 \end{align}||
However, there are two exponents which have certain special characteristics.
Exponent 0
Any base with the exponent |0| is equal to |1.|
|
||2^0=1|| |
||\left(\dfrac{1}{5}\right)^0=1|| |
||a^0=1|| |
Exponent 1
Any base with the exponent |1| remains unchanged.
|
||2^1=2|| |
||\left(\frac{1}{5}\right)^1 =\frac{1}{5}|| |
||a^1=a|| |
After mastering the concept of exponents, it becomes clear that the notation can have different results depending on the set of numbers used. In the case of real numbers, two forms of writing are generally used: decimal notation and fractional notation. Exponential notation can also be useful when working with square and cubic numbers and when using the laws of exponents.
||a^n = \underbrace{a \times a \times a \times ... \times a}_{n \ \text{times}}||
||\text{where }\ a \in \mathbb{R}\quad \text{and}\quad n \in \mathbb{N}^*||
Thus, the exponent refers to the number of times the base must be multiplied by itself.
Example 1
What is the result (power) of |7^5|?
||\begin{align} \color{red}{7}^\color{blue}{5} &= \underbrace{\color{red}{7 \times 7 \times 7 \times 7 \times 7}}_{\color{blue}{5 \ \text{times}}}\\\\ &=16 \ 807 \end{align}||
Example 2
What is the result (power) of |\text{-}1{.}5^4|?
||\begin{align} \text{-}\color{red}{1{.}5}^\color{blue}{4}&= \text{-}(\color{red}{1{.}5})^4 \\ &= \text{-}(\underbrace{\color{red}{1{.}5 \times 1{.}5 \times 1{.}5\times 1{.}5}}_{\color{blue}{4 \ \text{times}}})\\ &=\text{-}(5{.}062 \ 5) \\ &= \text{-} 5{.}062 \ 5\end{align}||
Example 3
What is the result (power) of |(\text{-}1{.}5)^4|?
||\begin{align} (\color{red}{\text{-}1{.}5})^\color{blue}{4} &= \underbrace{\color{red}{(\text{-}1{.}5) \times (\text{-}1{.}5) \times (\text{-}1{.}5) \times (\text{-}1{.}5)}}_{\color{blue}{4 \ \text{times}}}\\ &=5{.}062 \ 5\end{align}||
In the section above, we note that the base of Example 2 is |1{.}5| while in Example 3, the base is |\text{-}1{.}5|. We notice that the result is more or less the same; only the sign changes. To understand the difference between these two examples, it is important to look at the parentheses and the number of times the base is multiplied by itself. The rule below will help you to predict the sign of a power.
When a base is negative
Example: |(\text{-}9)^n|
-
When the exponent |n| is an even number, the negative base is multiplied by itself an even number of times, so the power will be positive.
-
When the exponent |n| is an odd number, the negative base is multiplied by itself an odd number of times, so the power will be negative.
When a base is positive
Example: |\text{-}(\text{9}^n)|
-
Regardless of the exponent’s parity, the answer will be negative.
Moreover, with decimal notation, the multiplication results in the final answer generally having more decimal places than the original base. Whenever possible, to ensure precision, transform the decimal notation into fractional notation and then perform the necessary calculations.
In addition, exponential notation can also be used with algebraic expressions.
Exponentiation of algebraic expressions requires algebraic concepts and exponentiation. Thus, when performing addition or subtraction with this notation, it is crucial to adhere to the exact definition of exponentiation:
||\begin{align} \color{red}{(x-3)}^\color{blue}{2} &= \underbrace{\color{red}{(x-3) \times (x-3)}}_{\color{blue}{2 \ \text{times}}}&&\text{expansion according to the definition}\\\\ &= x^2-3x - 3x + 9&& \text{double distribution}\\ &=x^2-6x+9 &&\text{grouping like terms}\end{align}||
Regardless of the type of base, the negative exponent always has the same impact on the base: it is necessary to take the reciprocal. Observe the sign of the base and the number of times it is multiplied by itself.
Natural number in fractional notation
|| a = \frac{a}{1}||
||\text{where}\ a \in \mathbb{N}^*||
Definition
||\begin{align} a^{\text{-}n} &= \left(\frac{a}{1}\right)^{\text{-}n}\\\\ &= \left(\frac{1}{a}\right)^n\end{align}||
||\text{where }\ a \in \mathbb{R}^* \quad \text{and}\quad n \in \mathbb{N}^*||
When the base is a real number written in decimal notation, pay close attention that the rules for writing fractions are respected.
Example 1
What is the result (power) of |3^{\text{-}3}|?
\begin{align} \color{red}{3}^\color{blue}{\text{-}3} &=\left(\frac{1}{\color{red}{3}}\right)^{\color{blue}{3}} && \text{definition of a negative exponent} \\\\ &= \underbrace{\frac{1}{\color{red}{3}} \times \frac{1}{\color{red}{3}}\times \frac{1}{\color{red}{3}}}_{\color{blue}{3 \ \text{times}}}&& \text{definition of exponentiation}\\\\ &=\frac{1}{3 \times 3 \times 3}\\\\ &= \frac{1}{27}\end{align}
Example 2
What is the result (power) of |(\text{-}5)^{\text{-}4}|?
\begin{align} (\color{red}{\text{-}5})^\color{blue}{\text{-}4} &=\left(\frac{1}{\color{red}{\text{-}5}}\right)^{\color{blue}{4}} && \text{definition of a negative exponent} \\\\ &= \underbrace{\frac{1}{\color{red}{\text{-}5}} \times \frac{1}{\color{red}{\text{-}5}}\times \frac{1}{\color{red}{\text{-}5}} \times \frac{1}{\color{red}{\text{-}5}}}_{\color{blue}{4 \ \text{times}}}&& \text{definition of exponentiation}\\\\ &=\frac{1}{\text{-}5 \times \text{-}5 \times \text{-}5 \times \text{-}5}\\\\ &= \frac{1}{625}\end{align}
Example 3
What is the result (power) of |0{.}3^{\text{-}3}|?
\begin{align}\color{red}{0{.}3}^\color{blue}{\text{-}3} &=\left(\frac{1}{\color{red}{0{.}3}}\right)^{\color{blue}{3}} && \text{definition of a negative exponent} \\\\ &=\left(\frac{10}{\color{red}{3}}\right)^{\color{blue}{3}} && \text{fraction-writing convention}\\\\ &=\underbrace{\left(\frac{10}{\color{red}{3}} \times \frac{10}{\color{red}{3}}\times \frac{10}{\color{red}{3}}\right)}_{\color{blue}{3 \ \text{times}}}&& \text{definition of exponentiation}\\\\ &=\left(\frac{10 \times 10 \times 10}{3 \times 3 \times 3}\right)\\\\ &=\frac{1\ 000}{27}\end{align}
Example 4
What is the result (power) of |\text{-}(1{.}5)^{\text{-}4}|?
\begin{align} \text{-}(\color{red}{1{.}5})^\color{blue}{\text{-}4} &=\text{-}\left(\frac{1}{\color{red}{1{.}5}}\right)^{\color{blue}{4}} && \text{definition of a negative exponent} \\\\ &=\text{-}\left(\frac{10}{\color{red}{15}}\right)^{\color{blue}{4}} && \text{fraction-writing convention}\\\\ &=\text{-}\ \underbrace{\left(\frac{10}{\color{red}{15}} \times \frac{10}{\color{red}{15}}\times \frac{10}{\color{red}{15}} \times \frac{10}{\color{red}{15}}\right)}_{\color{blue}{4 \ \text{times}}}&& \text{definition of exponentiation}\\\\ &=\text{-}\left(\frac{10\times 10 \times 10 \times 10}{15 \times 15 \times 15 \times 15}\right)\\\\ &=\text{-}\frac{10 \ 000}{50\ 625}\\\\ &=\text{-}\frac{16}{81} &&\text{fraction simplification} \end{align}
To understand the reason why we take the reciprocral of the base when it has a negative exponent, observe the following:
Each time the exponent decreases by |1,| the result must be divided by the value of the base, that is, |4.| So when going from |0| to |-1,| divide |1| by |4.| This results in |\dfrac{1}{4}.|
By understanding the relationship between the power and the value of the base, it can be deduced that |4| and |\dfrac{1}{4}| are the inverse, or reciprocral, of each other.
In terms of writing in exponential notation, a special notation exists when the base |10| is used.
More precisely, writing a number with the base |10| refers to writing in scientific notation.
In this section, fractional exponents can be converted by applying a square root.
||a^{\frac{m}{n}} = \sqrt[n]{a^m}||
||\text{where }\ a \in \mathbb{R},\quad m \in \mathbb{Z},\quad \text{and}\quad n \in \mathbb{N}^*||
Naturally, an exponent can have more than one special characteristic.
Example 1
What is the result (power) of |16^{\frac{1}{2}}|? ||\begin{align} \color{red}{16}^{\frac{\color{magenta}{1}}{\color{blue}{2}}} &=\sqrt[\color{blue}{2}]{\color{red}{16}^\color{magenta}{1}} && \text{definition of a fractional exponent} \\ &= \sqrt{16} && \\ &= \pm 4\end{align}||
Example 2
What is the result (power) of |(\text{-}16)^{\frac{1}{2}}|? ||\begin{align} (\color{red}{\text{-}16})^{\frac{\color{magenta}{1}}{\color{blue}{2}}} &=\sqrt[\color{blue}{2}]{\color{red}{(\text{-}16})^\color{magenta}{1}} && \text{definition of a fractional exponent} \\ &= \sqrt{\text{-}16} && \\ &= \emptyset \end{align}||
Example 3
What is the result (power) of |8^{\frac{\text{-}2}{3}}|? ||\begin{align} \color{red}{8}^{\frac{\color{magenta}{\text{-}2}}{\color{blue}{3}}} &= \left(\frac{1}{\color{red}{8}}\right)^{\frac{\color{magenta}{2}}{\color{blue}{3}}}&& \text{definition of a negative exponent}\\\\ &=\sqrt[\color{blue}{3}]{\left(\frac{1}{\color{red}{8}}\right)^\color{magenta}{2}} && \text{definition of a fractional exponent} \\\\ &=\sqrt[\color{blue}{3}]{\frac{1}{8}\times \frac{1}{8}} && 2^\text{nd} \text{definition of an exponent} \\\\
&= \sqrt[3]{\frac{1}{64}} && \\\\ &= \frac{1}{4}\end{align}||
Example 4
What is the result (power) of |\text{-}3{.}28^{\frac{2}{3}}|? ||\begin{align} \text{-}\color{red}{3{.}28}^{\frac{\color{magenta}{2}}{\color{blue}{3}}} &= \text{-}\sqrt[\color{blue}{3}]{\color{red}{3{.}28}^\color{magenta}{2}} && \text{definition of a fractional exponent} \\\\ &=\text{-}\sqrt[\color{blue}{3}]{3{.}28\times 3{.}28} && 2^\text{nd} \text{definition of an exponent} \\\\ &\approx \text{-}\sqrt[3]{10{.}758} && \\\\ &\approx\text{-} 2{.}21\end{align}||
In Example 2, there are numbers that are part of the integers (|\mathbb{Z}|) for which it is impossible to calculate the square root.
By definition, a square root, if it exists, can be positive or negative.
|\sqrt{25}| means "Which number when multiplied by itself, results in |25|?"
So two results are possible:
||\begin{align}5 \times 5 &= 25\\\\ \text{or}\\\\ (\text{-}5) \times (\text{-}5) &= 25\end{align}||
So,
||\sqrt{25}= \pm 5||
On the other hand, in many contexts, the negative answer is often rejected since it cannot be used for the value of a length or time for example.
The only downside to calculating fractional exponents with decimal numbers is the loss of precision. Switch from decimal notation to fractional notation and then make the appropriate calculations to ensure precision. To learn how to do this, see the next section
||\begin{align}\left(\frac{a}{b}\right)^n &= \underbrace{\frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \times ... \times \frac{a}{b}}_{n \ \text{times}}\\ &= \frac{a^n}{b^n}\end{align}||
||\text{where }\ {a,\ b} \in \mathbb{R}^* \quad \text{and}\quad n \in \mathbb{N}^*||
However, brackets are just as important in exponential notation where the base is in fractional form.
|\dfrac{a}{b}^n = \dfrac{(a^n)}{b}\quad| while |\quad\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}|
Using the definitions and properties of exponents, the power of a fraction is actually equal to the division of two powers, since the exponent over each element of the fraction must be distributed.
Example 1
What is the result (power) of |\left(\dfrac{2}{3}\right)^3|?
||\begin{align} \left(\color{red}{\frac{2}{3}}\right)^\color{blue}{3} &= \frac{2^3}{3^3}\\\\ &= \frac{8}{27}\end{align}||
Example 2
What is the result (power) of |\text{-}\left(\dfrac{5}{4}\right)^4|?
||\begin{align} \text{-}\left(\color{red}{\frac{5}{4}}\right)^\color{blue}{4} &= \text{-}\left(\frac{5^4}{4^4}\right)\\\\&=\text{-}\left(\frac{625}{256}\right) \\\\ &=\text{-} \frac{625}{256}\end{align}||
Example 3
What is the result (power) of |\left(\text{-}\dfrac{\sqrt{2}}{3}\right)^2|?
||\begin{align} \left(\color{red}{\text{-}\frac{\sqrt{2}}{3}}\right)^\color{blue}{2} &=\frac{(\text{-}\sqrt{2})^2}{3^2}\\\\&=\frac{2}{9}\end{align}||
||\begin{align}\left(\frac{\color{red}{a}}{\color{magenta}{b}}\right)^{\text{-}n} &= \left(\frac{\color{magenta}{b}}{\color{red}{a}}\right)^n\\\\ &= \frac{\color{magenta}{b}^n}{\color{red}{a}^n}\end{align}||
||\text{where }\ {\color{red}{a},\color{magenta}{b}} \in \mathbb{R}^* \quad \text{and}\quad n \in \mathbb{N}^*||
Once again, the definitions and properties of exponents simplify the process and calculations.
Example 1
What is the result (power) of |\text{-}\left(\dfrac{\color{red}{6}}{\color{magenta}{5}}\right)^{\text{-}2}|? ||\begin{align} \text{-}\left(\frac{\color{red}{6}}{\color{magenta}{5}}\right)^\color{blue}{\text{-}2} &=\text{-}\left(\frac{\color{magenta}{5}}{\color{red}{6}}\right)^{\color{blue}{2}} && \text{definition of a negative exponent} \\\\ &=\text{-}\left(\frac{\color{magenta}{5}^\color{blue}{2}}{\color{red}{6}^\color{blue}{2}}\right)&&\text{definition of an exponent}\\\\ &= \text{-}\left(\frac{25}{36}\right) \\\\ &= \frac{\text{-}25}{36}\end{align}||
Example 2
What is the result (power) of |\left(\text{-}\dfrac{\color{red}{3}}{\color{magenta}{7}}\right)^{\text{-}3}|? ||\begin{align} \left(\text{-}\frac{\color{red}{3}}{\color{magenta}{7}}\right)^\color{blue}{\text{-}3} &=\left(\text{-}\frac{\color{magenta}{7}}{\color{red}{3}}\right)^{\color{blue}{3}} && \text{definition of a negative exponent} \\\\ &=\frac{(\text{-}\color{magenta}{7})^\color{blue}{3}}{\color{red}{3}^\color{blue}{3}}&&\text{definition of an exponent}\\\\ &= \frac{\text{-}343}{27}\end{align}||
||\begin{align}\left(\frac{a}{b}\right)^{\frac{m}{n}} &= \sqrt[n]{\left(\frac{a}{b}\right)^m}\\\\ &= \frac{\sqrt[n]{a^m}}{\sqrt[n]{b^m}}\end{align}||
||\text{where }\ {a,\ b}\in \mathbb{R}^*,\quad m \in \mathbb{Z},\quad \text{and}\quad n \in \mathbb{N}^*||
Example 1
What is the result (power) of |\left(\text{-}\dfrac{1}{27}\right)^{\frac{1}{3}}|? ||\begin{align} \left( \color{red}{\text{-}\frac{1}{27}}\right)^{\frac{\color{magenta}{1}}{\color{blue}{3}}} &=\sqrt[\color{blue}{3}]{\color{red}{\left(\text{-}\frac{1}{27}\right)}^\color{magenta}{1}} && \text{definition of a fractional exponent} \\\\ &= \frac{\sqrt[\color{blue}{3}]{(\text{-}1)^\color{magenta}{1}}}{\sqrt[\color{blue}{3}]{27^\color{magenta}{1}}} && \text{distribution of roots}\\\\ &= \frac{\text{-}1}{3} \end{align}||
Pour valider ta compréhension à propos de l'exponentiation et des lois des exposants de façon interactive, consulte la MiniRécup suivante :
