10.4 Divide monomials  (Page 2/5)

Simplify:

1. ⓐ $\frac{a^5}{a^9}$
2. ⓑ $\frac{x^{11}}{x^7}$

Solution

ⓐ
Since 9>5, there are more $a$ 's in the denominator and so we will end up with factors in the denominator.	$\frac{a^5}{a^9}$
Use the Quotient Property for $n>m,\frac{a^m}{a^n}=\frac{1}{a^{n-m}}.$
Simplify.	$\frac{1}{a^4}$

ⓑ
Notice there are more factors of $x$ in the numerator, since 11>7. So we will end up with factors in the numerator.	$\frac{x^{11}}{x^7}$
Use the Quotient Property for $m>n,\frac{a^m}{a^n}=a^{n-m}.$
Simplify.	$x^4$

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Simplify expressions with zero exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\frac{a^m}{a^m}.$ From earlier work with fractions, we know that

In words, a number divided by itself is $1.$ So $\frac{x}{x}=1,$ for any $x$ ( $x\ne 0$ ), since any number divided by itself is $1.$

The Quotient Property of Exponents shows us how to simplify $\frac{a^m}{a^n}$ when $m>n$ and when $n<m$ by subtracting exponents. What if $m=n$ ?

Consider first $\frac{8}{8},$ which we know is $1.$

Now we will simplify $\frac{a^m}{a^m}$ in two ways to lead us to the definition of the zero exponent    .

In this text, we assume any variable that we raise to the zero power is not zero.

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let's look at ${\left(2x\right)}^0.$ We can use the product to a power rule to rewrite this expression.

This tells us that any non-zero expression raised to the zero power is one.

Simplify expressions using the quotient to a power property

Now we will look at an example that will lead us to the Quotient to a Power Property.

Notice that the exponent applies to both the numerator and the denominator.

We see that ${\left(\frac{x}{y}\right)}^3$ is $\frac{x^3}{y^3}.$

$\begin{array}{ccccc}\text{We write:}\hfill & & & & {\left(\frac{x}{y}\right)}^3\hfill \\ & & & & \frac{x^3}{y^3}\hfill \end{array}$

This leads to the Quotient to a Power Property for Exponents.

An example with numbers may help you understand this property: