About Quotient of Powers Property
What is meant by powers in math?
In mathematics, powers designate expressions that represent the repeated multiplication of the same factor. \(4\times 4\), for example, can be written as \(4^2\), where 4 is the base and 2 is the exponent.
The value of 5 to the second power is \(5^2\), also known as 5 square.
Exponents
The exponent of a number indicates the number of times it has been multiplied by itself. \(3^4\), for example, denotes a four-fold multiplication of 3. \(3\times 3 \times 3\times 3\) is its expanded form of \(3^4\). Exponent is another name for the power of a number. It could be a whole number, a fraction, a negative number, or even a decimal.
Difference between Powers and Exponents
Power | Exponent |
|---|---|
The multiplication of the same number is represented by the expression. | It is the number of times the base number is multiplied by itself and then used as a factor. |
It symbolizes the base and the exponent. | It represents the quantity that describes the power with which a number is multiplied. |
The base 2 in \(2^3=2\times2\times2\) is multiplied three times and can be 2 to the power 3, or 2 to the third power. | In \(2^3=2\times2\times2\) is the exponent that represents the number of times the value will be multiplied with 2 as the base. |
Powers can be multiplied when the base is the same. | Exponents are added when powers are multiplied. |
Quotient of Powers Property
The quotient of powers property states that we can subtract the powers of the same base. This can be done when the same base with different powers are divided by one another. We can observe the subtraction of powers from the general expression,
\(\frac{x^a}{x^b}=x^{a-b}\) where \(x\neq 0\)
Solved Quotient of Power Property Examples
Example 1: Simplify \(\frac{5^{14}}{5^4}\)
Solution:
Since the bases are the same in the division problem, the exponents are subtracted.
\(\frac{5^{14}}{5^4}\)\(=5^{14 – 4}\) Quotient of powers property
\(=5^{10}\) Simplify
Example 2: Simplify \(\frac{\left(x^7\right)\left(y^9\right)}{xy^2}\)
Solution:
This example includes two bases, but the top and bottom bases are identical. Simply match the bases and use the Quotient of Powers Property to solve the problem.
\(\frac{\left(x^7\right)\left(y^9\right)}{xy^2}\)
\(=x^{7-1}y^{9-2}\) Quotient of powers property
\(=x^6y^7\) Simplify
Example 3: Simplify \(\frac{\left(x^6\right)\left(y^5\right)}{xy^3}\)
Solution:
This example includes two bases, but the top and bottom bases are identical. Simply match the bases and use the property of the quotient of powers to solve the problem.
\(\frac{\left(x^6\right)\left(y^5\right)}{xy^3}\)
\(=x^{6-1}y^{5-3}\) Quotient of powers property
\(=x^5y^2\) Simplify
Example 4: Tennessee is expected to have a population of \(6⋅5.9^8\) people in 2040. Calculate the average population per square mile in Tennessee in 2040.
Solution:
Divide the projected population of Tennessee in 2040 by the land area to get the average number of people per square mile in 2040.
People per square mile \(\ =\frac{\left(\text{Population in}\ 2040\right)}{\text{Land area}}\)
\(=\frac{6\cdot{5.9}^8}{{5.9}^6}\) Substitute
\(=6\cdot\frac{{5.9}^8}{{5.9}^6}\) Rewrite
\(=6\cdot{5.9}^2\) Quotient of powers property
\(=208.86\) Evaluate
So, in 2040, Tennessee’s population density is expected to be around 208.86 people per square mile.
Example 5: Iceland has a population of approximately \(\ 2.94\ \times\ {10}^5\) people that are located in an area that covers \(1.03\ \times\ {10}^5\) square miles approximately. How many people are there per square mile, given these conditions?
Solution:
To find the people who are living per square mile, divide the population by the land area of Iceland.
So, the number of people per square mile = \(\frac{\text{Population}}{\text{Land area} }=\frac{2.94\times{10}^5}{1.03\times{10}^5}\)
\(=\frac{2.94}{1.03}\times{10}^{5-5}\) Quotient of powers property
\(\approx2.85\times{10}^0\) Simplify
\(\approx2.85\)
Hence, there are approximately 3 people per square mile in Iceland.
A number’s power (or exponent) determines how many times it can be multiplied. It’s written as a small number or in technical terms the “superscript” to the right of the base number for which the result of the power needs to be calculated.
An exponent is a number that appears in front of a number as a superscript. In other words, it denotes that the base has been “boosted to a certain level of strength”. Exponents are also known by other names, such as index and power. If is a positive number and is the exponent, \(m^{n}\) denotes that m has been multiplied by itself n times.
The following are some examples of exponents:
- \(5\times 5\times7\times 7\times 7=5^2\times 7^3\)
- \(-8\times -8\times -8\times -8=(-8)^4\)
- \(x\times x\times x\times x\times x=x^5\)