The
quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as
$\text{\hspace{0.17em}}\frac{{y}^{m}}{{y}^{n}},$ where
$\text{\hspace{0.17em}}m>n.\text{\hspace{0.17em}}$ Consider the example
$\text{\hspace{0.17em}}\frac{{y}^{9}}{{y}^{5}}.\text{\hspace{0.17em}}$ Perform the division by canceling common factors.
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.
$\frac{{y}^{9}}{{y}^{5}}={y}^{9-5}={y}^{4}$
For the time being, we must be aware of the condition
$\text{\hspace{0.17em}}m>n.\text{\hspace{0.17em}}$ Otherwise, the difference
$\text{\hspace{0.17em}}m-n\text{\hspace{0.17em}}$ could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.
The quotient rule of exponents
For any real number
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and natural numbers
$\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}n,$ such that
$\text{\hspace{0.17em}}m>n,$ the quotient rule of exponents states that
$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$
Using the quotient rule
Write each of the following products with a single base. Do not simplify further.
Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the
power rule of exponents . Consider the expression
$\text{\hspace{0.17em}}{\left({x}^{2}\right)}^{3}.\text{\hspace{0.17em}}$ The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.
The exponent of the answer is the product of the exponents:
$\text{\hspace{0.17em}}{\left({x}^{2}\right)}^{3}={x}^{2\cdot 3}={x}^{6}.\text{\hspace{0.17em}}$ In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.
${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.
For any real number
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and positive integers
$\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}n,$ the power rule of exponents states that
${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$
Using the power rule
Write each of the following products with a single base. Do not simplify further.
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication