10.4 Divide Monomials

Learning objectives

Learning objectives

• Simplify expressions using the Quotient Property of Exponents

• Simplify expressions with zero exponents

• Simplify expressions using the Quotient to a Power Property

• Simplify expressions by applying several properties

• Divide monomials

Simplify Expressions Using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.

As before, we'll try to discover a property by looking at some examples.

$\begin{array}{lccccclccc} \mathrm{Consider} & & & \frac{{x}^{5}}{{x}^{2}} & & & \mathrm{and} & & & \frac{{x}^{2}}{{x}^{3}} \\ \mathrm{What do they mean?} & & & \frac{x\cdot{}x\cdot{}x\cdot{}x\cdot{}x}{x\cdot{}x} & & & & & & \frac{x\cdot{}x}{x\cdot{}x\cdot{}x} \\ \mathrm{Use the Equivalent Fractions Property.} & & & \frac{x\cdot{}x\cdot{}x\cdot{}x\cdot{}x}{x\cdot{}x\cdot{}1} & & & & & & \frac{x\cdot{}x\cdot{}1}{x\cdot{}x\cdot{}x} \\ \mathrm{Simplify.} & & & {x}^{3} & & & & & & \frac{1}{x} \\ \end{array}$

Notice that in each case the bases were the same and we subtracted the exponents.

• When the larger exponent was in the numerator, we were left with factors in the numerator and $1$ in the denominator, which we simplified.

• When the larger exponent was in the denominator, we were left with factors in the denominator, and $1$ in the numerator, which could not be simplified.

We write:

A couple of examples with numbers may help to verify this property.

When we work with numbers and the exponent is less than or equal to $3,$ we will apply the exponent. When the exponent is greater than $3$, we leave the answer in exponential form.

Example 10.45

Solution

1. Apply the quotient property. For powers with the same base, subtract the denominator exponent from the numerator exponent.
2. Simplify part ⓐ. The exponent difference is $10-8=2$, so $\frac{x^{10}}{x^8}=x^2$.
3. Simplify part ⓑ. The exponent difference is $9-2=7$, so $\frac{2^9}{2^2}=2^7$.

Takeaway: For a quotient with the same base, subtract the denominator exponent from the numerator exponent, so $x^{10}/x^8=x^2$ and $2^9/2^2=2^7$.

Example 10.46

Solution

1. Compare the exponents in each quotient. A larger denominator exponent leaves a negative exponent difference, which must be rewritten as a reciprocal.
2. Simplify part ⓐ. The difference $10-15=-5$ gives $b^{-5}=\frac{1}{b^5}$.
3. Simplify part ⓑ. The difference $3-5=-2$ gives $3^{-2}=\frac{1}{3^2}=\frac{1}{9}$.

Takeaway: When the denominator has the larger exponent in a same-base quotient, the result is $1$ divided by the remaining base power, giving $1/b^5$ and $1/9$.

Example 10.47

Solution

1. Apply the quotient property to part ⓐ. Since both powers have base $a$, subtract the denominator exponent from the numerator exponent: $\frac{{a}^{5}}{{a}^{9}}=a^{5-9}=a^{-4}$.
2. Rewrite the negative exponent as a reciprocal. Using $a^{-n}=\frac{1}{a^n}$ gives $\frac{1}{a^4}$. The first figure shows this reciprocal form as $\frac{1}{a^{9-5}}$.
3. Apply the quotient property to part ⓑ. The common base is $x$, so $\frac{{x}^{11}}{{x}^{7}}=x^{11-7}=x^4$. The second figure displays the exponent difference $11-7$.
4. State both results. Part ⓐ is $\frac{1}{a^4}$ and part ⓑ is $x^4$.

Takeaway: When dividing equal bases, subtract the denominator exponent and interpret any negative result by placing the base in the denominator.

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 $m=n$?

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

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

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

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

Example 10.48

Solution

1. State the zero-exponent rule. Every nonzero number raised to the zero power equals $1$, regardless of the base or its magnitude.
2. Simplify part ⓐ. Since $12$ is nonzero, $12^0=1$ regardless of whether the base is greater or less than $1$.
3. Simplify part ⓑ. Under the usual assumption that $y\ne0$, $y^0=1$, so both expressions simplify to $1$.

Takeaway: Every nonzero base raised to the zero power equals 1, so $12^0=1$ and $y^0=1$.

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 ${(2x)}^{0}.$ 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.

Example 10.49

Solution

1. Identify the complete base. The zero exponent applies to the entire nonzero product $7z$, not merely to $z$.
2. Apply the zero-exponent rule. The expression $(7z)^0$ therefore has value $1$.
3. State the simplified result. No coefficient or variable remains because the whole base was raised to the zero power.

Takeaway: Any nonzero expression raised to the zero power equals 1, so $(7z)^0=1$.

Example 10.50

Solution

1. Locate each zero exponent. In part ⓐ, the exponent applies to the entire product; in part ⓑ, it applies only to $y$.
2. Simplify part ⓐ. The nonzero product ${(-3{x}^{2}y)}^{0}$ has value $1$ by the zero-exponent rule.
3. Simplify part ⓑ. Replace $y^0$ with $1$, giving $-3x^2\cdot{}1=-3x^2$.
4. Compare the results. The whole base becomes $1$ in part ⓐ, while only $y$ becomes $1$ in part ⓑ.

Takeaway: A zero exponent applies only to its own base, so $(-3x^2y)^0=1$ while $-3x^2y^0=-3x^2$.

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 ${(\frac{x}{y})}^{3}$ is $\frac{{x}^{3}}{{y}^{3}}.$

$\begin{array}{lcccl} \mathrm{We write:} & & & & {(\frac{x}{y})}^{3} \\ & & & & \frac{{x}^{3}}{{y}^{3}} \\ \end{array}$

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

An example with numbers may help you understand this property:

Example 10.51

Solution

1. Apply the quotient to a power property to part ⓐ. Raise the numerator and denominator to the second power: $\left(\frac{5}{8}\right)^2=\frac{5^2}{8^2}$. The first two figures show the original expression and this separated form, with both exponents highlighted.
2. Simplify the numerical powers. Since $5^2=25$ and $8^2=64$, the fraction becomes $\frac{25}{64}$, the result shown in the third figure.
3. Apply the same property to part ⓑ. Raise both parts to the fourth power: $\left(\frac{x}{3}\right)^4=\frac{x^4}{3^4}$. The fourth and fifth figures display this original and separated forms with the exponents $4$ highlighted.
4. Evaluate the denominator. Since $3^4=81$, part ⓑ simplifies to $\frac{x^4}{81}$, as shown in the sixth figure.
5. Apply the property to part ⓒ and state the results. Raising both parts to the third power gives $\frac{y^3}{m^3}$, with the final two figures showing that separated form and its highlighted exponents. Therefore the three answers are $\frac{25}{64}$, $\frac{x^4}{81}$, and $\frac{y^3}{m^3}$.

Takeaway: Raising a quotient to a power raises the numerator and denominator separately to that same power.

Simplify Expressions by Applying Several Properties

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Example 10.52

Solution

1. Simplify the nested power in the numerator. Multiplying exponents gives ${({x}^{2})}^{3}=x^{2\cdot{}3}=x^6$.
2. Rewrite the quotient. The expression becomes $\frac{x^6}{x^5}$, with matching bases in the numerator and denominator.
3. Subtract the exponents. The quotient property gives $x^{6-5}=x^1=x$, so the simplified result is $x$.

Takeaway: Simplify the power in the numerator before dividing, so $\frac{({x}^{2})^{3}}{x^{5}}=\frac{x^{6}}{x^{5}}=x$.

Example 10.53

Solution

1. Simplify the nested power in the denominator. The power property gives ${({m}^{2})}^{4}=m^{2\cdot{}4}=m^8$ by multiplying the inner and outer exponents.
2. Rewrite the quotient with equal powers. The expression becomes $\frac{m^8}{m^8}=m^{8-8}$ by subtracting the denominator exponent.
3. Apply the zero-exponent rule. The remaining exponent is $0$, and $m^0=1$ for nonzero $m$, so the result is $1$.

Takeaway: Multiplying exponents can make the numerator and denominator identical, and $\frac{m^{8}}{({m}^{2})^{4}}=\frac{m^{8}}{m^{8}}=m^{0}=1$.

Example 10.54

Solution

1. Simplify inside the parentheses first. The matching bases allow the quotient property to give $\frac{x^7}{x^3}=x^{7-3}=x^4$.
2. Apply the outer square. The expression becomes ${(x^4)}^{2}$ after the inner quotient is simplified.
3. Multiply the exponents. The power property gives ${(x^4)}^{2}=x^{4\cdot{}2}=x^8$.
4. State the result. The simplified expression is $x^8$.

Takeaway: Simplify inside the parentheses before applying the outer exponent, so $(\frac{x^{7}}{x^{3}})^{2}=(x^{4})^{2}=x^{8}$.

Example 10.55

Solution

1. Distribute the outer power to the fraction. The quotient to a power property gives ${(\frac{p^2}{q^5})}^{3}=\frac{{(p^2)}^{3}}{{(q^5)}^{3}}$.
2. Multiply the exponents inside each power. The numerator becomes $p^{2\cdot{}3}=p^6$, and the denominator becomes $q^{5\cdot{}3}=q^{15}$.
3. State the simplified fraction. Since the bases differ, no further cancellation is possible, so the result is $\frac{p^6}{q^{15}}$.

Takeaway: Raise the numerator and denominator separately when a fraction is powered, so $(\frac{p^{2}}{q^{5}})^{3}=\frac{p^{6}}{q^{15}}$.

Example 10.56

Solution

1. Apply the quotient to a power property. Raising the fraction to the fourth power gives $\frac{{(2{x}^{3})}^{4}}{{(3y)}^{4}}$.
2. Distribute the fourth power inside the fraction. The numerator and denominator become $\frac{2^4({x^3})^4}{3^4y^4}$.
3. Simplify the numerical powers. The coefficients give $2^4=16$ and $3^4=81$.
4. Simplify the variable powers. Multiplying exponents gives $(x^3)^4=x^{12}$ and $y^4=y^4$, so the result is $\frac{16x^{12}}{81y^4}$.

Takeaway: A power applies to every factor in a fraction, so $(\frac{2x^{3}}{3y})^{4}=\frac{16x^{12}}{81y^{4}}$.

Example 10.57

Solution

1. Simplify each nested power. The numerator exponents become $2\cdot{}3=6$ and $2\cdot{}4=8$, while the denominator exponent becomes $5\cdot{}4=20$.
2. Combine the numerator powers. The same base $y$ appears in both numerator factors, so adding exponents gives $y^6y^8=y^{14}$.
3. Apply the quotient property. The expression becomes $\frac{y^{14}}{y^{20}}=y^{14-20}=y^{-6}$.
4. Rewrite the negative exponent. Using the negative-exponent rule gives $y^{-6}=\frac{1}{y^6}$, which is the final result.

Takeaway: Combine powers of the same base before dividing, so $\frac{({y}^{2})^{3}({y}^{2})^{4}}{({y}^{5})^{4}}=\frac{y^{14}}{y^{20}}=\frac{1}{y^{6}}$.

Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.

Example 10.58

Solution

1. Rewrite division as a fraction. The expression $56{x}^{5}\div7{x}^{2}$ becomes $\frac{56{x}^{5}}{7{x}^{2}}$.
2. Separate numerical and variable factors. This gives $\frac{56}{7}\cdot\frac{x^5}{x^2}$.
3. Divide the coefficients. The coefficient quotient is $\frac{56}{7}=8$.
4. Subtract the exponents. The quotient property gives $x^{5-2}=x^3$, so the complete quotient is $8{x}^{3}$.

Takeaway: Rewrite division as a fraction, then divide coefficients and subtract exponents of like bases, giving $56x^{5}\div7x^{2}=8x^{3}$.

When we divide monomials with more than one variable, we write one fraction for each variable.

Example 10.59

Solution

1. Separate the quotient into like factors. Rewrite the expression as $\frac{42}{-7}\cdot\frac{x^2}{x}\cdot\frac{y^3}{y^5}$.
2. Divide the coefficients. The quotient $\frac{42}{-7}$ is $-6$, so the sign is negative.
3. Subtract each variable exponent. The powers of $x$ give $x^{2-1}=x$, and the powers of $y$ give $y^{3-5}=y^{-2}$.
4. Rewrite the negative exponent. Since $y^{-2}=\frac{1}{y^2}$, the simplified quotient is $-\frac{6x}{y^2}$.

Takeaway: In a monomial quotient, divide coefficients and subtract each variable's denominator exponent from its numerator exponent, giving $-\frac{6x}{y^{2}}$.

Example 10.60

Solution

1. Separate the quotient into like factors. Rewrite the expression as $\frac{24}{48}\cdot\frac{a^5}{a}\cdot\frac{b^3}{b^4}$.
2. Simplify the coefficient quotient. The fraction $\frac{24}{48}$ reduces to $\frac{1}{2}$.
3. Subtract each variable exponent. The powers of $a$ give $a^{5-1}=a^4$, and the powers of $b$ give $b^{3-4}=b^{-1}$.
4. Place the negative power in the denominator. Since $b^{-1}=\frac{1}{b}$, the final quotient is $\frac{a^4}{2b}$.

Takeaway: In a monomial quotient, divide coefficients and subtract each variable's denominator exponent from its numerator exponent, giving $\frac{a^{4}}{2b}$.

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example 10.61

Solution

1. Reduce the numerical coefficients. Dividing $14$ and $21$ by their common factor $7$ gives $\frac{2}{3}$.
2. Subtract the exponent of $x$. The exponent difference $7-11=-4$ gives the factor $\frac{1}{x^4}$.
3. Subtract the exponent of $y$. The exponent difference $12-6=6$ gives the factor $y^6$.
4. Combine the simplified factors. Putting the reciprocal and positive power together yields $\frac{2y^6}{3x^4}$.

Takeaway: Dividing coefficients and subtracting exponents independently gives $\frac{14x^{7}y^{12}}{21x^{11}y^{6}}=\frac{2y^{6}}{3x^{4}}$.

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.

Example 10.62

Solution

1. Simplify the numerator first. The fraction bar groups the two monomials, so multiply their coefficients and add like exponents to get $30{x}^{5}{y}^{5}$.
2. Rewrite the whole fraction. The expression becomes $\frac{30{x}^{5}{y}^{5}}{6{x}^{4}{y}^{5}}$.
3. Cancel the common variable powers. Dividing by $6x^4y^5$ leaves the coefficient $5$ and the power $x^{5-4}=x$.
4. State the quotient. The common $y^5$ factors also cancel, so the simplified result is $5x$.

Takeaway: The fraction bar is a grouping symbol, so simplify the numerator first: $(3x^{3}y^{2})(10x^{2}y^{3})=30x^{5}y^{5}$, and dividing by $6x^{4}y^{5}$ leaves $5x$.

Section 10.4 Exercises

Section 10.4 Exercises

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Divide Monomials

In the following exercises, divide the monomials.

Mixed Practice

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Simplify:

1. ⓐ $\,\frac{{x}^{12}}{{x}^{9}}$

2. ⓑ $\,\frac{{7}^{14}}{{7}^{5}}$

2. Simplify:

1. ⓐ $\,\frac{{y}^{23}}{{y}^{17}}$

2. ⓑ $\,\frac{{8}^{15}}{{8}^{7}}$

3. Simplify:

1. ⓐ $\,\frac{{x}^{8}}{{x}^{15}}$

2. ⓑ $\,\frac{{12}^{11}}{{12}^{21}}$

4. Simplify:

1. ⓐ $\,\frac{{m}^{17}}{{m}^{26}}$

2. ⓑ $\,\frac{{7}^{8}}{{7}^{14}}$

5. Simplify:

1. ⓐ $\,\frac{{b}^{19}}{{b}^{11}}$

2. ⓑ $\,\frac{{z}^{5}}{{z}^{11}}$

1. Simplify:

1. ⓐ $\,\frac{{p}^{9}}{{p}^{17}}$

2. ⓑ $\,\frac{{w}^{13}}{{w}^{9}}$

2. Simplify:

1. ⓐ $\,{17}^{0}$

2. ⓑ $\,{m}^{0}$

3. Simplify:

1. ⓐ $\,{k}^{0}$

2. ⓑ $\,{29}^{0}$

4. Simplify:

1. ⓐ $\,{(7{x}^{2}y)}^{0}$

2. ⓑ $\,7{x}^{2}{y}^{0}$

5. Simplify:

1. ⓐ $\,-23{x}^{2}{y}^{0}$

2. ⓑ $\,{(-23{x}^{2}y)}^{0}$

1. Simplify:

1. ⓐ $\,{(\frac{7}{9})}^{2}$

2. ⓑ $\,{(\frac{y}{8})}^{3}$

3. ⓒ $\,{(\frac{p}{q})}^{6}$

2. Simplify:

1. ⓐ $\,{(\frac{1}{8})}^{2}$

2. ⓑ $\,{(\frac{-5}{m})}^{3}$

3. ⓒ $\,{(\frac{r}{s})}^{4}$

3. Simplify: $\frac{{({a}^{4})}^{5}}{{a}^{9}}.$

4. Simplify: $\frac{{({b}^{5})}^{6}}{{b}^{11}}.$

5. Simplify: $\frac{{k}^{11}}{{({k}^{3})}^{3}}.$

1. Simplify: $\frac{{d}^{23}}{{({d}^{4})}^{6}}.$

2. Simplify: ${(\frac{{f}^{14}}{{f}^{8}})}^{2}.$

3. Simplify: ${(\frac{{b}^{6}}{{b}^{11}})}^{2}.$

4. Simplify: ${(\frac{{m}^{3}}{{n}^{8}})}^{5}.$

5. Simplify: ${(\frac{{t}^{10}}{{u}^{7}})}^{2}.$

1. Simplify: ${(\frac{5b}{9{c}^{3}})}^{2}.$

2. Simplify: ${(\frac{4{p}^{4}}{7{q}^{5}})}^{3}.$

3. Simplify: $\frac{{({y}^{4})}^{4}{({y}^{3})}^{5}}{{({y}^{7})}^{6}}.$

4. Simplify: $\frac{{(3{x}^{4})}^{2}{({x}^{3})}^{4}}{{({x}^{5})}^{3}}.$

5. Find the quotient: $63{x}^{8}\mathrm{÷}9{x}^{4}.$

1. Find the quotient: $96{y}^{11}\mathrm{÷}6{y}^{8}.$

2. Find the quotient: $\frac{-84{x}^{8}{y}^{3}}{7{x}^{10}{y}^{2}}.$

3. Find the quotient: $\frac{-72{a}^{4}{b}^{5}}{-8{a}^{9}{b}^{5}}.$

4. Find the quotient: $\frac{16{a}^{7}{b}^{6}}{24a{b}^{8}}.$

5. Find the quotient: $\frac{27{p}^{4}{q}^{7}}{-45{p}^{12}{q}^{}}.$

1. Find the quotient: $\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}.$

2. Find the quotient: $\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}.$

3. Find the quotient: $\frac{(3{x}^{4}{y}^{5})(8{x}^{2}{y}^{5})}{12{x}^{5}{y}^{8}}.$

4. Find the quotient: $\frac{(-6{a}^{6}{b}^{9})(-8{a}^{5}{b}^{8})}{-12{a}^{10}{b}^{12}}.$

5. $\frac{{4}^{8}}{{4}^{2}}$

1. $\frac{{x}^{12}}{{x}^{3}}$

2. $\frac{{r}^{5}}{r}$

3. $\frac{{y}^{4}}{{y}^{20}}$

4. $\frac{{10}^{3}}{{10}^{15}}$

5. $\frac{a}{{a}^{9}}$

1. $-{7}^{0}$

2. 1. ⓐ $\,{(10p)}^{0}$

2. ⓑ $\,10{p}^{0}$

3. 1. ⓐ $\,{(-27{x}^{5}y)}^{0}$

2. ⓑ $\,-27{x}^{5}{y}^{0}$

4. 1. ⓐ $\,{15}^{0}$

2. ⓑ $\,{15}^{1}$

5. ${(\frac{3}{2})}^{5}$

1. ${(\frac{m}{6})}^{3}$

2. ${(\frac{x}{y})}^{10}$

3. ${(\frac{a}{3b})}^{2}$

4. $\frac{{({x}^{2})}^{4}}{{x}^{5}}$

5. $\frac{{({u}^{3})}^{4}}{{u}^{10}}$

1. $\frac{{y}^{8}}{{({y}^{5})}^{2}}$

2. ${(\frac{{x}^{2}}{{x}^{8}})}^{3}$

3. ${(\frac{{a}^{4}\,\cdot\,{a}^{6}}{{a}^{3}})}^{2}$

4. $\frac{{({y}^{3})}^{5}}{{({y}^{4})}^{3}}$

5. $\frac{{({x}^{3})}^{6}}{{({x}^{4})}^{7}}$

1. ${(\frac{2{r}^{3}}{5s})}^{4}$

2. $\frac{{({r}^{2})}^{5}\,{({r}^{4})}^{2}}{{({r}^{3})}^{7}}$

3. $\frac{{(3{x}^{4})}^{3}\,{(2{x}^{3})}^{2}}{{(6{x}^{5})}^{2}}$

4. $48{b}^{8}\mathrm{÷}6{b}^{2}$

5. $36{x}^{3}\mathrm{÷}(-2{x}^{9})$

1. $\frac{18{x}^{3}}{9{x}^{2}}$

2. $\frac{-35{x}^{7}}{-42{x}^{13}}$

3. $\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}$

4. $\frac{8m{n}^{10}}{64m{n}^{4}}$

5. $\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}$

1. $\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}$

2. $\frac{(6{m}^{2}n)(5{m}^{4}{n}^{3})}{3{m}^{10}{n}^{2}}$

3. $\frac{(4{u}^{5}{v}^{4})(15{u}^{8}v)}{(12{u}^{3}v)({u}^{6}v)}$

4. 1. ⓐ $\,15{n}^{10}+3{n}^{10}$

2. ⓑ $\,15{n}^{10}-3{n}^{10}$

3. ⓒ $\,15{n}^{10}\cdot{}3{n}^{10}$

4. ⓓ $\,15{n}^{10}\mathrm{÷}3{n}^{10}$

5. 1. ⓐ $\,{q}^{5}\cdot{q}^{3}$

2. ⓑ $\,{({q}^{5})}^{3}$

1. 1. ⓐ $\,\frac{{z}^{6}}{{z}^{5}}$

2. ⓑ $\,\frac{{z}^{5}}{{z}^{6}}$

2. $(4y)(12{y}^{7})\mathrm{÷}8{y}^{2}$

3. $\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}$

4. $\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}$

5. $\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}$

1. Memory One megabyte is approximately ${10}^{6}$ bytes. One terabyte is approximately ${10}^{12}$ bytes. How many megabytes are in one terabyte?

2. Mai simplifies the quotient $\frac{{y}^{3}}{y}$ by writing $\frac{{y}^{3}}{y}=3.$ What is wrong with her reasoning?

3. Roxie thinks ${n}^{0}$ simplifies to $0.$ What would you say to convince Roxie she is wrong?