## Math 8 Chapter 1 Lesson 10: Dividing a monomial by a monomial

## 1. Theory

– To divide monomial A by monomial B (in case A is divisible by B), we do the following:

- Divide the coefficient of monomial A by the coefficient of monomial B.
- Divide the power of each variable in A by the power of the same variable in B.
- Multiply the results you just found together.

**– Note: **x≠0, m, n ϵ N, m ≥ n then:

- \({x^m}:{x^n} = {x^{m – n}}\) if m > n
- \({x^m}:{x^n} = 1\) if m = n

## 2. Illustrated exercise

**Question 1:** Calculate:

a. \({7^5}:{7^3}\)

b. \({16^2}:{( – 6)^2}\)

**Solution guide**

**Question a:**

\(\begin{array}{l} {7^5}:{7^3}\\ = {7^{5 – 3}}\\ = {7^2}\\ = 49 \end{array} \)

**Sentence b:**

\(\begin{array}{l} {16^2}:{( – 6)^2}\\ = {\left( {\frac{{ – 16}}{6}} \right)^2} \\ = {\left( {\frac{{ – 8}}{3}} \right)^2}\\ = \frac{{64}}{9} \end{array}\)

**Verse 2: **Monomial division

a. \({\left( { – x} \right)^7}:{\left( { – x} \right)^5}\)

b. \(\frac{5}{2}{x^5}{y^5}:\left( { – \frac{1}{2}{x^4}{y^4}} \right)\)

**Solution guide**

**Question a:**

\(\begin{array}{l} {\left( { – x} \right)^7}:{\left( { – x} \right)^5}\\ = {\left( x \right) ^{7 – 5}}\\ = {\left( x \right)^2}\\ = {x^2} \end{array}\)

**Sentence b:**

\(\begin{array}{l} \frac{5}{2}{x^5}{y^5}:\left( { – \frac{1}{2}{x^4}{y^) 4}} \right)\\ = \frac{5}{2}{\left( {xy} \right)^5}:\left( { – \frac{1}{2}} \right){\ left( {xy} \right)^4}\\ = – 5{\left( {xy} \right)^{5 – 4}}\\ = – 5xy \end{array}\)

**Question 3:** Calculate the value of the expression \(32{x^6}{y^5}{z^{10}}:8{x^4}{y^3}{z^{10}}\) with x = 3, y = 2, z = 1996

**Solution guide**

\(\begin{array}{l} 32{x^6}{y^5}{z^{10}}:8{x^4}{y^3}{z^{10}}\\ = 4{x^{6 – 4}}{y^{5 – 3}}{z^{10 – 10}}\\ = 4{x^2}{y^2}\\ = 4 \times {3 ^2} \times {2^2}\\ = 144 \end{array}\)

### 3.1. Essay exercises

**Question 1: **Calculate:

a. \({{13}^{6}}:{{13}^{4}}\)

b. \({{24}^{4}}:{{(-14)}^{4}}\)

**Verse 2:** Monomial division

a. \({{\left( -xy \right)}^{5}}:{{\left( -xy \right)}^{4}}\)

b. \(\frac{7}{3}{{x}^{6}}{{y}^{5}}:\left( -\frac{5}{3}{{x}^{4}} {{y}^{4}} \right)\)

**Question 3: **Calculates the value of the expression \(45{{x}^{7}}{{y}^{7}}{{z}^{9}}:9{{x}^{4}}{{y) }^{4}}{{z}^{9}}\) with x = 2, y = 1, z = 2020

## 3.2. Multiple choice exercises

**Question 1: **The result of dividing the monomial \(6x^3y^2z\) by the monomial \(2xyz\) gives us which of the following results?

A. \(3x^2y\)

B. \(3xyz\)

C. \(2xy^2z\)

D. \(2x^2y\)

**Verse 2: **The result of dividing the monomial \(5x^5y^3\) by the monomial \(3xy\) is

A. \(\frac{5}{3} xy\)

B. \(\frac{5}{3} x^4y^2\)

C. \(\frac{3}{5}xyz\)

D. \(x^2y^3z\)

**Question 3: **Divide \({x^{300}}\) by \(( – {x^{250}})\) we get which of the following results?

A. \(- {x^{550}}\)

B. \( {-x^{50}}\)

C. \({x^{ – 50}}\)

D. \({x^{50}}\)

**Question 4: **Calculate the value of the expression \(A=\left( {20{x^3}{y^4}{z^2}} \right):(5{x^2}{y^2}z)\ ) at x=2; y=3, z=1

A. A=72

B. A=27

C. A=48

D. A=24

**Question 5: **Performing division \({x^{n + 19}}:{x^{14}}\,\,\,\left( {n \in N} \right)\) we get the following result ?

A. \({x^{n – 5}}\,\)

B. \(\,\,{x^{n + 5}}\,\)

C. \({x^2}\)

D. \({x^5}\)

## 4. Conclusion

Through this lesson, you should achieve the following goals:

- Grasp the monomial.
- It is possible to divide a monomial by a monomial.

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