Do you know that numbers can be powered? Obviously not like powering a car or any machine but mathematically.
In this article, we will define exponents and their properties and more.
What is an exponent?
An exponent shows us the number of times a number is multiplied by itself. It is often regarded as either Powers.
The exponent \(2^3\) shows that \(2\) is to multiply itself \(3\) times:
\[2^3=2\times2\times2\]
Components of an exponent
An exponent contains two major parts: the part that stays at the top is called the power, and the part that is below which carries the power, is called the base.
In \(5^7\), \(7\) is the power or the exponent while \(5\) is the base.
Properties of exponents
In order to carry out operations of exponents, some rules have been laid down to guide you easily.
Multiplication property of exponents
The multiplication or product rule of exponents states that
The product of two or more numbers with the same base is equal to the common base to the power of the sum of the exponents
\[a^m\times a^n=a^{m+n}\]
Expand \(2^3\times 2^2\).
Solution:
\[2^3\times 2^2=2^{3+2}=2^5\]
Verification
\[2^3\times 2^2=(2\times 2\times 2)\times (2\times 2)=2^5\]
Expand \(a^4\times a^6\).
Solution:
\[a^4+a^6=a^{4+6}=a^{10}\]
Verification
\[a^4\times a^6=(a\times a\times a\times a)\times (a\times a\times a\times a\times a\times a)=a^{10}\]
Expand \(2^3\times 3^3 \times 2^4\).
Solution:
\[2^3\times 3^3\times 2^4=(2^3\times 2^4)\times 3^3=2^{3+4}\times 3^3=2^7\times 3^3\]
Division property of exponents
The division or quotient rule of exponents states that
The quotient of two or more numbers with the same base is equal to the common base to the power of the difference of the exponents
\[\dfrac{a^m}{a^n}=a^m\div a^n=a^{m-n}\]
A quotient of two or more numbers with different bases is equal to their direct division,
\[a^m\div b^n=\dfrac{a^m}{b^n}\]
Simplify \)\dfrac{2^3}{2}\).
Solution:
\[\dfrac{2^3}{2}=2^{3-1}=2^2\]
Verification
\[\dfrac{2^3}{2}=\dfrac{2\times 2\times 2}{2}=2\times 2=2^2\]
Simplify \(\dfrac{b^7}{b^3}\).
Solution:
\[\dfrac{b^7}{b^3}=b^{7-3}=b^4\]
Verification
\[\dfrac{b^7}{b^3}=\dfrac{b\times b\times b\times b\times b\times b\times b}{b\times b\times b}=b\times b\times b\times b=b^4\]
Simplify \(\dfrac{2^5\times 3^7}{3^4}\).
Solution:
\[\dfrac{2^5\times 3^7}{3^7}=2^5\times \dfrac{3^7}{3^4}=2^5\times 3^{7-4}=2^5\times 3^3\]
Zero exponent property
The null or zero exponent rule states that
Any nonzero number raised to the exponent of \(0\) is equal to \(1\). That is, for every \(a\neq 0\), we have \(a^0=1\).
Proof of the Zero exponent property
Using the division rule for exponents, for every \(a\neq 0\), we have
\[\dfrac{a}{a}=a^{1-1}=a^0\]
On the other hand, we have \(\dfrac{a}{a}=1\), thus
\[\dfrac{a}{a}=a^{1-1}=a^0=1\]
a. \(2^0\)
b. \(-2^0\)
c. \((-2)^0\)
Solution:
a. Anything to the power of \(0\) is equal to \(1\). So we have
\[2^0=1\]
b. Here, the base is \(2\), with a \(-1\) multiplied out front. It becomes
\[\begin{align}-2^0&=-1\times 2^0=\\&=-1\times 1=\\&=-1\end{align}\]
c. Here, the base is \(-2\), and anything to the power of \(0\) equals \(1\). It becomes
\[(-2)^0=1\]
Note how important the brackets are in the above examples \(b\) and \(c\).
Negative exponent property
The negative exponent property states that
A base with a negative exponent is equal to the reciprocal of the base raised to the opposite of the exponent
That is for every \(a\neq 0\), we have
\[a^{-m}=\dfrac{1}{a^m}\]
Proof of the negative exponent property
For every \(a\neq 0\), we have \(a^{-m}=a^{0-m}=\dfrac{a^0}{a^m}=\dfrac{1}{a^m}\).
\[2^{-1}=\dfrac{1}{2}\]
\[9^{-3}=\left(\dfrac{1}{9}\right)^3\]
\[6^{-5}=\left(\dfrac{1}{6}\right)^5\]
Properties of rational exponents
A number raised to a fraction exponent is equal to the denominator's root of the base raised to the exponent of the numerator, that is \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\).
In particular, \(a^{\frac{1}{n}}=\sqrt[n]{a}\).
Find the value of the following expressions.
a. \(125^{\frac{1}{3}}\)
b. \(2^{\frac{4}{3}}\)
c. \(16^{-\frac{3}{2}}\)
Solution:
a. The first step is to express \(125\) as a product of its prime factors,
\[125=5\times 5\times 5=5^3\]
Thus we have,
\[125^{\frac{1}{3}}=\sqrt[3]{125}=\sqrt[3]{5^3}=5\]
b.
\[2^{\frac{4}{3}}=\sqrt[3]{2^4}=\sqrt[3]{16}\]
c. There are two ways to solve this question and both ways involve using the fractional exponent rule and the negative exponent rule.
First, use the fractional exponent rule.
\[\begin{align}16^{-\frac{3}{2}}&=\left(\sqrt[2]{16}\right)^{-3}=\\&=(4)^{-3}\end{align}\]
From here, we use the negative exponent rule
\[\begin{align}(4)^{-3}&=\dfrac{1}{4^3}=\\&=\dfrac{1}{64}\end{align}\]
Solving it in an alternative way, you will first use the negative exponent rule.
\[16^{-\frac{3}{2}}=\dfrac{1}{16^{\frac{3}{2}}}\]
We will now use the fractional exponent rule on the denominator.
\[\begin{align}\dfrac{1}{16^{\frac{3}{2}}}&=\dfrac{1}{\left(\sqrt[2]{16}\right)^3}=\\&=\dfrac{1}{4^3}=\\&=\dfrac{1}{64}\end{align}\]
You get the same answer!
Power of a product property
The power of a product property states that
When a product of two numbers is raised to a power, the resulting answer is equal to the product of each number bearing that exponent separately.
In other words, the product of two different numbers with the same exponent is equal to the product of each of these numbers raised to its exponent, that is
\[(ab)^m=a^m\times b^m\]
We note that
\[(ab)^m=a^m\times b^m=b^m\times a^m=(ba)^m\]
Verify that \(6^3=2^3\times 3^3\).
Solution:
Method 1
On the one hand, we have; \(6^3=6\times 6\times 6=216\).
On the other hand, \(2^3\times 3^3=8\times 27=216\).
Thus \(6^3=2^3\times 3^3\).
Method 2
\[6^3=(2\times 3)^2=2^3\times 3^3\]
Power of a quotient property
The power of a quotient property states that
When a quotient of two numbers is raised to a power, the resulting answer is equal to the quotient of each number bearing that exponent separately
In other words, the quotient of two different numbers with the same exponent is equal to the quotient of each of these numbers raised to its exponent, that is
\[\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}\]
Verify that \(2^3=\dfrac{6^3}{3^3}\).
Solution:
Method 1
To begin,
\[2^3=2\times 2\times 2=8\]
Also,
\[\begin{align}\dfrac{6^3}{3^3}&=\dfrac{6\times 6\times 6}{3\times 3\times 3}=\\\\&=\dfrac{^2\cancel{6}\times ^2\cancel{6}\times ^2\cancel{6}}{^1\cancel{3}\times ^1\cancel{3}\times ^1\cancel{3}}=\\\\&= 2\times 2\times 2=\\\\&=8\end{align}\]
This implies that
\[2^3=\dfrac{6^3}{3^3}\]
Method 2
Apply the quotient property;
\[\begin{align}\dfrac{6^3}{3^3}&=\left(\dfrac{6}{3}\right)^3=\\&=2^3=\\&=8\end{align}\]
Therefore;
\[2^3=\dfrac{6^3}{3^3}\]
Power of a power property
A number raised to an exponent is raised to another exponent is equal to the number raised to the product of the exponents, that is
\[\left(a^m\right)^n=a^{m\times n}=a^{mn}=a^{nm}=a^{n\times m}=\left(a^n\right)^m\]
Verify that \(\left(2^3\right)^2=2^6\).
Solution:
Method 1
On the one hand, we have
\[2^3=2\times 2\times 2=8\]
Thus
\[\left(2^3\right)^2=8^2=64\]
On the other hand,
\[2^6=2^{3+3}=2^3\times 2^3=8\times 8=64\]
Thus,
\[\left(2^3\right)^2=2^6=64\]
Method 2
\[\left(2^3\right)^2=2^{3\times 2}=2^6=64\]
Examples of properties of exponents
Calculate the following without the use of calculators.
a. \((-3x^3y^2)(2x^6y^5)\)
b. \((2b)^{-4}\)
c. \(\left(\dfrac{-6x^6}{3x^3}\right)^{-2}\)d. \(81^{\frac{3}{4}}\)
e. \(\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}\)
Solution:
a. For the expression,
\[(-3x^3y^2)(2x^6y^5)\]
We express them as separate products,
\[(-3x^3y^2)(2x^6y^5)=(-3\times x^3\times y^2)\times (2\times x^6\times y^5)\]
We expand the brackets,
\[(-3x^3y^2)(2x^6y^5)=-3\times x^3\times y^2\times 2\times x^6\times y^5\]
Next, we bring like terms together,
\[\begin{align}(-3x^3y^2)(2x^6y^5)&=-3\times 2\times x^3\times x^6\times y^2\times y^5=\\&=-6\times \left(x^{3+6}\right)\times\left(y^{2+5}\right)=\\&=-6\times x^9\times y^7=\\&=-6x^9y^7\end{align}\]
b. For the expression,
\[(2b)^{-4}\]
We first get rid of the negative exponent, we apply the reciprocal rule,
\[(2b)^{-4}=\dfrac{1}{(2b)^4}=\dfrac{1}{2^4b^4}=\dfrac{1}{16b^4}\]c. For the expression,
\[\left(\dfrac{-6x^6}{3x^3}\right)^{-2}\]
To get rid of the negative exponent, we apply the reciprocal rule,
\[\left(\dfrac{-6x^6}{3x^3}\right)^{-2}=\left(\dfrac{3x^3}{-6x^6}\right)^2\]
We then divide the like terms of the expression in the bracket,
\[\left(\dfrac{-6x^6}{3x^3}\right)^{-2}=\left(\dfrac{3x^3}{-6x^6}\right)^2=\left(-\dfrac{1}{2}\left(x^{3-6}\right)\right)^2=\left(-\dfrac{1}{2}x^{-3}\right)^2\]
Afterward, we distribute the exponent \(2\) to the product inside the bracket to get,
\[\begin{align}\left(\dfrac{-6x^6}{3x^3}\right)^{-2}=\left(-\dfrac{1}{2}\times \dfrac{1}{x^3}\right)^2=\\&=\left(-\dfrac{1}{2x^3}\right)^2=\\&=\dfrac{(-1)^2}{(2x^3)^2}=\\&=\dfrac{1}{2^2(x^3)^2}=\\&=\dfrac{1}{4x^6}\end{align}\]
d. For the expression,
\[81^{\frac{3}{4}}\]
we recall first the fraction exponent rule,
\[81^{\frac{3}{4}}=\left(\sqrt[4]{81}\right)^3=\sqrt[4]{81^3}\]
But
\[81=9^2=\left(3^2\right)^2=3^4\]
Hence
\[81^{\frac{3}{4}}=\sqrt[4]{81^3}=\sqrt[4]{\left(3^4\right)^3}=\sqrt[4]{\left(3^3\right)^4}=3^3\]
We can see in another way, we recall that,
\[\left(a^m\right)=a^{mn}\]
Thus,
\[81^{\frac{3}{4}}=\left(3^4\right)^{\frac{3}{4}}=3^{4\times \frac{3}{4}}=3^3\]
e. For the expression,
\[\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}\]
We first expand the numerator,
\[\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}=\dfrac{-12\times m^4\times n^3\times m^3\times n^2}{36m^7n^5}\]
We next bring like terms together, to get
\[\begin{align}\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}&=\dfrac{-12\times m^4\times m^3\times n^3\times n^2}{36m^7n^5}=\\&=\dfrac{-12\times m^{4+3}\times n^{3+2}}{36m^7n^5}=\\&=\dfrac{-12\times m^7\times n^5}{36m^7n^5}=\\&=\dfrac{-12\times m^7\times n^5}{36\times m^7\times n^5}\end{align}\]
We then divide like terms to get,
\[\begin{align}\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}&=\left(\dfrac{-12}{36}\right)\times \left(\dfrac{m^7}{m^7}\right)\times\left(\dfrac{n^5}{n^5}\right)=\\&=\left(-\dfrac{1}{3}\right)\times m^{7-7}\times n^{5-5}=\\&=\left(-\dfrac{1}{3}\right)\times m^0\times n^0\end{align}\]
We recall that any nonzero number raised to the exponent \(0\) is \(1\), we get
\[\dfrac{-12m^4n^3(m^3n^2)}{36m^7n^5}=\left(-\dfrac{1}{3}\right)\times 1\times 1=-\dfrac{1}{3}\]
Simplify the expression \(\dfrac{m^{\frac{3}{4}}\times n^{\frac{1}{2}}}{m^{\frac{1}{2}}\times n^{-\frac{3}{2}}}\).
Solve when \(m= 16\) and \(n= 3\).
Solution:
\[\begin{align}\dfrac{m^{\frac{3}{4}}\times n^{\frac{1}{2}}}{m^{\frac{1}{2}}\times n^{-\frac{3}{2}}}&=\dfrac{m^{\frac{3}{4}}}{m^{\frac{1}{2}}}\times \dfrac{n^{\frac{1}{2}}}{n^{-\frac{3}{2}}}=\\&=\left(m^{\frac{3}{4}}\div m^{\frac{1}{2}}\right)\times \left(n^{\frac{1}{2}}\div n^{-\frac{3}{2}}\right)=\\&=m^{\frac{3}{4}-\frac{1}{2}}\times n^{\frac{1}{2}-\left(-\frac{3}{2}\right)}=\\&=m^{\frac{1}{4}}\times n^2\end{align}\]
Substitute the value of \(m\) as \(16\) and \(n\) as \(3\) into the expression;
\[\begin{align}m^{\frac{1}{4}}\times n^2&=16^{\frac{1}{4}}\times 3^2=\\&=\left(2^4\right)^{\frac{1}{4}}\times 3^2=\\&=2^{4\times \frac{1}{4}}\times 3^2=\\&=2\times 9=\\&=18\end{align}\]
Scientific notation
The way numbers are commonly written is referred to as standard notation. However, Scientific Notation represents figures using the format,
\[q\times 10^p\quad\text{for}\quad 1\leq q < 10\]
with \(p\) being an integer.
Convert \(38 000 000 000\) meters per second to Scientific Notation.
Solution:
The first step is to count from your left to right. We have the number \(38 000 000 000\).
We have \(38\) and \(9\) zeros to the right of it.
We recall the scientific notation,
\[q\times 10^p\]
\[1\leq q<10\]
where \(p\) is an integer. Thus
\[38 000 000 000=38\times 10^9\]
Now \(38\) should be written as \(q\times 10^p\) as well. Thus,
\[38=3.8\times 10\]
Now we replace in the initial expression to get,
\[38 000 000 000=3.8\times 10\times 10^9=3.8\times 10^{10}\]
The length and bread of a rectangular mark is \(2\, \text{mm}\) and \(6\, \text{mm}\) respectively, calculate the perimeter in kilometers leaving your answer in standard form.
Solution:
The perimeter of a rectangle is given as
\[\begin{align}\text{Perimeter of the mark}&=2\times\left(\text{length}+\text{breadth}\right)=\\&=2\times\left(2\,\text{mm}+6\,\text{mm}\right)=\\&=2\times 8\,\text{mm}=\\&=16\,\text{mm}\end{align}\]
Recall that;
\[1000\,\text{m}=1\text{ km}\]
\[100\text{ cm}=1\text{ m}\]
\[10\text{ mm}=1\text{ cm}\]
\[100\times 10\text{ mm}=1\text{ m}\]
\[1000\times 100\times 10 \text{ mm}=1\text{ km}\]
\[1\times 10^6\text{ mm}=1\text{ km}\]
\[\dfrac{1\times 10^6\text{ mm}}{10^6}=\dfrac{1\text{ km}}{10^6}\]
\[\dfrac{1\times \cancel{10^6}\text{ mm}}{\cancel{10^6}}=\dfrac{1\text{ km}}{10^6}\]
\[1\text{ mm}=\dfrac{1}{10^6}\text{ km}\]
Recall that;
\[a^{-1}=\dfrac{1}{a}\]
Thus;
\[\dfrac{1}{10^6}=10^{-6}\]
This means that;
\[1\text{ mm}=10^{-6}\text{ km}\]
So convert \(16\text{ mm}\) to \(\text{km}\);
\[1\text{ mm}=10^{-6}\text{ km}\]
\[1\text{ mm}\times 16=10^{-6}\text{ km}\times 16\]
\[16\text{ mm}=16\times 10^{-6}\text{ km}\]
Now \(16\) should be written as \(q\times 10^p\) as well. Thus,
\[16=1.6\times 10\]
Now we replace it in the initial expression to get;
\[\begin{align}16\times 10^{-6}\text{ km}&=1.6\times 10\times 10^{-6}\text{ km}=\\&=1.6\times 10^1\times 10^{-6}\text{ km}=\\&=1.6\times 10^{-6+1}\text{ km}=\\&=1.6\times 10^{-5}\text{ km}\end{align}\]
\[\text{Perimeter of the mark}=1.6\times 10^{-5}\text{ km}\]
Properties of Exponents - Key takeaways
- An exponent shows us the number of times a number is multiplied by itself.
- A power consists of two main parts, the base, and the exponent.
- Exponents are simplified using their properties.
- Scientific notation is an easier representation of numbers using the notation \(q\times10^p\) where \(p\) is an integer and \(1\leq q<10\).
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