# 6.5 Divide monomials  (Page 2/4)

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Simplify: $\frac{{x}^{18}}{{x}^{22}}$ $\frac{{12}^{15}}{{12}^{30}}.$

$\frac{1}{{x}^{4}}$ $\frac{1}{{12}^{15}}$

Simplify: $\frac{{m}^{7}}{{m}^{15}}$ $\frac{{9}^{8}}{{9}^{19}}.$

$\frac{1}{{m}^{8}}$ $\frac{1}{{9}^{11}}$

Notice the difference in the two previous examples:

• If we start with more factors in the numerator, we will end up with factors in the numerator.
• If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

Simplify: $\frac{{a}^{5}}{{a}^{9}}$ $\frac{{x}^{11}}{{x}^{7}}.$

## Solution

1. Is the exponent of $a$ larger in the numerator or denominator? Since 9>5, there are more $a\text{'}\text{s}$ in the denominator and so we will end up with factors in the denominator.
 Use the Quotient Property, $\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}$ . Simplify.
2. Notice there are more factors of $x$ in the numerator, since 11>7. So we will end up with factors in the numerator.
 Use the Quotient Property, $\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}$ . Simplify.

Simplify: $\frac{{b}^{19}}{{b}^{11}}$ $\frac{{z}^{5}}{{z}^{11}}.$

${b}^{8}$ $\frac{1}{{z}^{6}}$

Simplify: $\frac{{p}^{9}}{{p}^{17}}$ $\frac{{w}^{13}}{{w}^{9}}.$

$\frac{1}{{p}^{8}}$ ${w}^{4}$

## Simplify expressions with an exponent of zero

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 your earlier work with fractions, you know that:

$\frac{2}{2}=1\phantom{\rule{2em}{0ex}}\frac{17}{17}=1\phantom{\rule{2em}{0ex}}\frac{-43}{-43}=1$

In words, a number divided by itself is 1. So, $\frac{x}{x}=1$ , for any $x\phantom{\rule{0.2em}{0ex}}\left(x\ne 0\right)$ , since any number divided by itself is 1.

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

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

$\begin{array}{cccccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{8}{8}& =\hfill & 1\hfill \\ \text{Write 8 as}\phantom{\rule{0.2em}{0ex}}{2}^{3}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{2}^{3}}{{2}^{3}}& =\hfill & 1\hfill \\ \text{Subtract exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{3-3}& =\hfill & 1\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{0}& =\hfill & 1\hfill \end{array}$

Now we will simplify $\frac{{a}^{m}}{{a}^{m}}$ in two ways to lead us to the definition of the zero exponent. In general, for $a\ne 0$ :

We see $\frac{{a}^{m}}{{a}^{m}}$ simplifies to ${a}^{0}$ and to 1. So ${a}^{0}=1$ .

## Zero exponent

If $a$ is a non-zero number, then ${a}^{0}=1$ .

Any nonzero number raised to the zero power is 1.

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

Simplify: ${9}^{0}$ ${n}^{0}.$

## Solution

The definition says any non-zero number raised to the zero power is 1.

1. $\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{9}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

2. $\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{n}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

Simplify: ${15}^{0}$ ${m}^{0}.$

1 1

Simplify: ${k}^{0}$ ${29}^{0}.$

1 1

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.

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{0}\hfill \\ \text{Use the product to a power rule.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{0}{x}^{0}\hfill \\ \text{Use the zero exponent property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1·1\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

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

Simplify: ${\left(5b\right)}^{0}$ ${\left(-4{a}^{2}b\right)}^{0}.$

## Solution

1. $\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(5b\right)}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

2. $\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(-4{a}^{2}b\right)}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

Simplify: ${\left(11z\right)}^{0}$ ${\left(-11p{q}^{3}\right)}^{0}.$

$1$ $1$

Simplify: ${\left(-6d\right)}^{0}$ ${\left(-8{m}^{2}{n}^{3}\right)}^{0}.$

$1$ $1$

## 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.

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{x}{y}\right)}^{3}\hfill \\ \text{This means:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{x}{y}·\frac{x}{y}·\frac{x}{y}\hfill \\ \text{Multiply the fractions.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{x·x·x}{y·y·y}\hfill \\ \text{Write with exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{x}^{3}}{{y}^{3}}\hfill \end{array}$

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

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

$\begin{array}{cccc}\text{We write:}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{x}{y}\right)}^{3}\hfill \\ & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{x}^{3}}{{y}^{3}}\hfill \end{array}$

-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her$66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
x > $110,000 bruce greater than$110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost$4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use? nawal Reply$1.38 worth of strawberries + $1.14 worth of blueberries which=$2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times$2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60 Melissa good Robinson 👍 Leitha thanks Melissa. Leitha nawal let's do another😊 Leitha we can't use emojis...I see now Leitha Sorry for the multi post. My phone glitches. Melissa Vina has$4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs$3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound? Vonna Reply 33.336 Robinson Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be$10 per square foot?
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have? mikayla Reply 2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75 Melissa John has$175 in $5 and$10 bills in his drawer. The number of $5 bills is three times the number of$10 bills. How many of each are in the drawer?
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the$4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total \$23,500? (Round your answer to the nearest tenth of a percent.)
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert