5.03 Quotient rule
Quotient rule
Exploration
Expand each of the expressions for the given values:
| Expression | Values | Substitute values | Expanded form | Simplified exponential form |
|---|---|---|---|---|
| \dfrac{2^{m}}{2^{n}} | {m=5},\,{n=3} | |||
| \dfrac{3^{s}}{3^{t}} | {s=4},\,{t=2} | |||
| \dfrac{4^{x}}{4^{y}} | {x=6},\,{y=1} |
- What do you notice about the relationship between the exponents in the simplified form and the original exponents?
What happens to the base when you divide two expressions with the same base?
If we wanted to simplify the expression a^{6} \div a^{2}, we could write it as:
We can see that common factors are divided out of the expanded expression, leaving a^4.
Consider the expression a^{6} \div a^{2}, which can be written as \dfrac{a^6}{a^2}.
Let's think about what this would look like if we expanded the expression:
\dfrac{a^6}{a^2} = \dfrac{a\cdot a \cdot a \cdot a \cdot a \cdot a}{a\cdot a}
\dfrac{a^6}{a^2} = \dfrac{a\cdot a \cdot a \cdot a }{1} \cdot \dfrac{a \cdot a}{a \cdot a}
\dfrac{a^6}{a^2} = \dfrac{a^4}{1} \cdot 1= a^4
We can avoid having to write each expression in expanded form by using the quotient rule: \dfrac{a^{m}}{a^{n}}=a^{m-n} where a is any base number, and m and n are powers.
That is, when dividing terms with a common base:
Keep the same base
Find the difference in the power.
Of course, we can also write this rule in the form a^{m}\div a^{n} = a^{m-n}
When using the quotient rule for exponents, the coefficients are handled separately from the exponents. Let's take a look at an example.
\dfrac{12x^5}{2x^2}
Divide the numerical coefficients: 12\div 2 =6
Apply the quotient rule to the exponents: \dfrac{x^5}{x^2} = x^{5-2}=x^3
The final result is:\dfrac{12x^5}{2x^2} = 6x^3
Examples
Example 1
Simplify the following expressions:
\dfrac{z^{14}}{z^{3}}
\dfrac{6 m^{9} n^{7}}{2 m^{4} n^{5}}
The quotient rule for exponents:
That is, when dividing terms with a common base:
Keep the same base
Find the difference in the power.
Power of a quotient rule
Exploration
Consider the mathematical expression \left(\dfrac{a}{b}\right)^n
Complete the table below by writing the expanded forms of the expressions and then simplifying your expressions:
| Expression | Expanded form | Simplify as the quotient of two powers |
|---|---|---|
| \left(\dfrac{2}{3} \right) ^2 | \dfrac{2^⬚}{3^⬚} | |
| \left(\dfrac{4}{5} \right)^3 | ||
| \left(\dfrac{6}{5} \right)^4 | ||
| \left(\dfrac{3}{4} \right)^2 | ||
| \left(\dfrac{5}{2} \right)^3 |
Can you identify any patterns or relationships between the original expression and the expanded form?
- Can you write a rule based on your observations?
When the power is applied to the entire quotient, we use the power of a power rule to write an expression for the power of a quotient rule as:\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}
Consider the expression \left(\dfrac{a}{b}\right)^5
This can be expanded as \left(\dfrac{a}{b}\right)^5 = \left(\dfrac{a\cdot a \cdot a \cdot a \cdot a}{b\cdot b\cdot b\cdot b\cdot b}\right) = \dfrac{(a^5)}{(b^5)} = \dfrac{a^5}{b^5}. Keep in mind we cannot simplify this any further because a and b are different bases.
Examples
Example 2
Simplify the following expressions:
\left(\dfrac{2x^6}{3y^3}\right)^4
\left( \dfrac{a^6 b^4}{a^2 b} \right) ^{3}
When we raise a fraction to a power, we can use the power of a quotient rule: \left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}