# Multiplying and Dividing Monomials

Monomials are polynomials with only one term. To multiply and divide monomials, use exponent’s rules.

## Step by step guide to Multiply and Divide Monomials

• When you divide two monomials you need to divide their coefficients and then divide their variables.
• In case of exponents with the same base, you need to subtract their powers.
• Exponent’s rules:
$$\color{ blue }{x^a×x^b=x^{a+b}}$$ , $$\color{ blue }{\frac{x^a}{x^b} =x^{a-b}}$$
$$\color{ blue }{\frac{1}{x^b} =x^{-b}}, \color{ blue }{(x^a)^b=x^{a×b}}$$
$$\color{ blue }{(xy)^a= x^a× y^a }$$

### Example 1:

Multiply expressions. $$(8x^5 )(-2x^4 )=$$

Solution:

Use multiplication property of exponents: $$\color{blue}{x^a×x^b=x^{a+b}} →x^5×x^4=x^9$$
Then: $$(8x^5 )(-2x^4 )=-16x^9$$

### Example 2:

Divide. $$\frac{-12x^4 y^3}{2xy^2 }=$$

Solution:

Use division property of exponents: $$\color{blue}{\frac{x^a}{x^b} =x^{a-b}} , \frac{x^4}{x}= x^{4-1}=x^3$$ and $$\frac{y^3}{y^2} =y$$
Then: $$\frac{-12x^4 y^3}{2xy^2 }=-6x^3 y$$

### Example 3:

Multiply expressions. $$(-3x^7 )(4x^3 )=$$

Solution:

Use multiplication property of exponents: $$\color{blue}{x^a×x^b=x^{a+b}} →x^7×x^3=x^{10 }$$
Then: $$(-3x^7 )(4x^3 )=-12x^{10}$$

### Example 4:

Dividing expressions. $$\frac{18x^2 y^5}{2xy^4}=$$

Solution:

Use division property of exponents: $$\color{blue}{\frac{x^a}{x^b} =x^{a-b}} , \frac{x^2}{x}= x^{2-1}=x$$ and $$\frac{y^5}{y^4} =y^{5-4}=y$$
Then: $$\frac{18x^2 y^5}{2xy^4}=9xy$$

## Exercises

### Simplify.

1. $$\color{blue}{(7x^4y^6)(4x^3y^4)}$$
2. $$\color{blue}{(15x^4) (3x^9)}$$
3. $$\color{blue}{(12x^2y^9)(7x^9y^{12})} \\\$$
4. $$\color{blue}{\frac{80x^{12 }y^9}{10x^6 y^7}} \\\$$
5. $$\color{blue}{\frac{95x^{18 }y^7}{5x^9 y^2}} \\\$$
6. $$\color{blue}{\frac{200x^3 y^8}{40x^3 y^7}} \\\$$

1. $$\color{blue}{28x^7y^{10}}$$
2. $$\color{blue}{45x^{13}}$$
3. $$\color{blue}{84x^{11}y^{21}}$$
4. $$\color{blue}{8x^6y^2}$$
5. $$\color{blue}{19x^9y^5}$$
6. $$\color{blue}{5y}$$