In this chapter we will learn different case of division of powers and the method to solve them with examples.
The concept is important as it would help you solve complex algebraic problems in higher classes.
Before starting the chapter, you should have basic understanding of the concepts of exponents and its terminologies. Click the red link to read about the same.
Exponent Basics
Let us first revise the basics of exponent.
When a number is multiplied by itself more than once, then we use the exponents to express the number.
For example;
Let number 3 is multiplied by itself four times.
\mathtt{\Longrightarrow \ 3\times 3\times 3\times 3}
Using exponents the expression can be written as \mathtt{3^{4}}
The big number at the center is called base. It tells the number which is being multiplied.
The small number at the top right corner above base is called power/exponent. It tells the number of times the multiplication is done.
In the above example;
3 ⟹ is the base
4 ⟹ is the exponent
To concept of base & power is very important to understand the different cases of division of exponents.
Methods of dividing exponents
There are two broad cases of division of exponents;
(a) Division of exponent when base is integer
(b) Division of exponents when base is in fraction form
We will discuss both the cases in detail.
Dividing exponents with integer base
In this topic, three sub – cases are possible;
(a) dividing numbers with base
(b) dividing numbers with different base but same power
(c) different base and different powers
Let us understand each of the topic in detail.
Dividing exponents with same base
When the numbers have same base, the simplification can be done by simply subtracting the powers while keeping the base same.
Here the division is expressed as;
\mathtt{a^{m} \div \ a^{n} \ =\ a^{m\ -\ n}}
This can also be written as;
\mathtt{\frac{a^{m}}{a^{n}} =a^{m\ -\ n}}
Let us see some examples for better clarity.
Example 01
Simplify \mathtt{\frac{2^{8}}{2^{3}}}
Solution
Note that both the dividing numbers have same base “2”.
The division can be simplified by simply subtracting the exponents.
\mathtt{\Longrightarrow \ \frac{2^{8}}{2^{3}}}\\\ \\ \mathtt{\Longrightarrow 2^{8\ -\ 3}}\\\ \\ \mathtt{\Longrightarrow \ 2^{5}}
Example 02
Simplify \mathtt{\frac{15^{10}}{15^{-3}}}
Solution
Again both the numbers have same base. Simplify by subtracting the powers.
\mathtt{\Longrightarrow \ \frac{15^{10}}{15^{-3}}}\\\ \\ \mathtt{\Longrightarrow 15^{10\ -\ ( -3)}}\\\ \\ \mathtt{\Longrightarrow 15^{10\ +3}}\\\ \\ \mathtt{\Longrightarrow \ 15^{13}}
Example 03
Simplify \mathtt{\frac{17^{-9}}{17^{-21}}}
Solution
\mathtt{\Longrightarrow \ \frac{17^{-9}}{17^{-21}}}\\\ \\ \mathtt{\Longrightarrow \ 17^{-9\ -\ ( -21)}}\\\ \\ \mathtt{\Longrightarrow \ 17^{-9\ +\ 21}}\\\ \\ \mathtt{\Longrightarrow \ 17^{12}}
Dividing exponents have different base but same power
When the exponents have different base and same power, the division can be simplified by dividing the base separately and then giving it the same exponent.
The above rule is expressed as follows;
\mathtt{\frac{a^{m}}{b^{m}} =\ \left(\frac{a}{b}\right)^{m}}
The expression can also be written as;
\mathtt{a^{m} \div \ b^{m} \ =\ ( a\div b)^{m\ }}
Let us see some examples related to this concept.
Example 01
Simplify the expression \mathtt{\frac{5^{3}}{4^{3}}}
Solution
Note that both the numbers have same power.
\mathtt{\Longrightarrow \ \frac{5^{3}}{4^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{5}{4}\right)^{3}}
Example 02
Simplify \mathtt{\frac{12^{5}}{36^{5}}}
Solution
\mathtt{\Longrightarrow \ \frac{12^{5}}{36^{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{12}{36}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{12}}{\cancel{36} \ 4}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{1}{4}\right)^{5}}
Dividing exponents with different base and power
In this case, we have to first find the value of number individually and then divide the number.
For example;
Simplify \mathtt{\frac{2^{4}}{3^{2}}}
Solution
Note that both the numbers have different base and powers.
So find the value of each number individually and then divide.
\mathtt{\Longrightarrow \ \frac{2^{4}}{3^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2\times 2\times 2\times 2}{3\times 3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{16}{9}}
Hence, the above expression has been simplified to 16/9 .
Example 02
Simplify \mathtt{\frac{12^{2}}{6^{3}}}
Solution
Note that both the numbers have different base and powers.
So find the value of individual exponents and then divide.
\mathtt{\Longrightarrow \ \frac{12^{2}}{6^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{12} \ \mathbf{2} \ \times \cancel{12} \ \mathbf{2}}{\cancel{6} \times \cancel{6} \times 6}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{6}}
Hence, the above division has been reduced to fraction 4/6.
Dividing exponents with Fraction base
Here the method is exactly similar to the division of integer base exponents (already discussed above).
The only difference is the presence of fraction number as base.
Here also, we have three types of division;
(a) Division of numbers with same base.
\mathtt{\ \left(\frac{a}{b}\right)^{m} \div \ \left(\frac{a}{b}\right)^{n} =\ \left(\frac{a}{b}\right)^{m-n}}
In this case also, we will simply subtract the powers to find the solution.
(b) Division of numbers with different base & same power.
\mathtt{\left(\frac{a}{b}\right)^{m} \div \ \left(\frac{c}{d}\right)^{m} =\ \left(\frac{a}{b} \div \frac{c}{d}\right)^{m}}
In this case, we can individually divide the base by keeping the same power.
(c) Different base and power
\mathtt{\left(\frac{a}{b}\right)^{m} \div \ \left(\frac{c}{d}\right)^{n} =\ \left(\frac{a}{b}\right)^{m} \div \left(\frac{c}{d}\right)^{n}}
Here we first have to individually find the value of each exponent and then do the division.
Conclusion
The division of fraction exponents is exactly same as the method of division of integer exponents.