2.5 Exponent Laws: Part 2

Leanne Thompson

2.5 Exponent Laws: Part 2

In this lesson, we will practice using the exponent laws to simplify more complicated expressions.

Simplifying Expressions with Coefficients

Example 1

Simplify 3m³ $\bullet$ 4m⁵.

Table 2.5.1
Steps	Solution
Rearrange the expression to put coefficients with coefficients and variables with variables.	3m³ $\bullet$ 4m⁵ (3)(4)(m³)(m⁵)
Multiply the coefficients, and use the Product Law to simplify the variables.	12m⁸

Practice 1

Simplify.

a) (12x³)(4x⁵)

b) (–5m⁹)(–5m⁸)

c) p³q²p⁴q

d) –2n³ $\times$ 5n⁶ $\times$ 2n

e) 3x²y³ $\bullet$ 2x⁴y² $\bullet$ x⁵y²

Example 2

Simplify $\frac{14a^5}{7a^2}$

Table 2.5.2
Steps	Solution
Rewrite the expression to put the coefficients together as one fraction and the variables together as one fraction.	$\frac{14a^5}{7a^2}$ $\left(\frac{14}{7}\right)$ $\left(\frac{a^5}{a^2}\right)$
Divide the coefficients, and use the Quotient Law to simplify the variables.	2a³

Practice 2

Simplify.

a) $\frac{12b^9}{3b^4}$

b) $\frac{-24z^3}{-8z}$

c) $\frac{-48x^5}{72x^4}$

d) $\frac{a^6b^6}{a^4b^3}$

e) $\frac{15s^7t^2}{20s^3t^2}$

Simplifying Expressions with Two or More Exponent Laws

To simplify some expressions, you will need to use two or more exponent laws.

Practice 3

Simplify.

a) (4p⁵)²

b) (–3x⁴y⁵)³

c) $\left(\frac{m^4}{n}\right)^3$

d) $\left(\frac{10a^4}{2b^5}\right)^2$

Practice 4

Simplify.

a) –7x³y⁶(2x²y²)⁴

b) (3r²s⁴)²(4r³s⁵)²

c) $\frac{(2m^4n^4)^3}{(mn^2)^3}$

d) $\frac{(5a^3b^6)^2}{5a^3b^4}$

Practice 5

Simplify.

a) (2a²a⁴b⁵)³(–5b⁴b²)²

b) $\left(\frac{36r^5s^6}{18r^3s}\right)^3$

c) $\left(\frac{-3w^7s^2 \bullet 6ws^4}{9w^5}\right)^2$

Homework

Simplify.

a) $\left(-8{u}^{6}\right)\left(-9u\right)$

b) $\left(4{a}^{3}b\right)\left(9{a}^{2}{b}^{6}\right)$

c) $\left(\dfrac{4}{7}r{s}^{2}\right)\left(14r{s}^{3}\right)$

d) $\left(\dfrac{2}{3}{x}^{2}y\right)\left(\dfrac{3}{4}x{y}^{2}\right)$

e) ${\left(8{a}^{3}\right)}^{2}{\left(2a\right)}^{4}$

f) ${\left(10{p}^{4}\right)}^{3}{\left(5{p}^{6}\right)}^{2}$

g) ${\left(\dfrac{1}{2}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2}$

h) ${\left(3{m}^{2}n\right)}^{2}{\left(2m{n}^{5}\right)}^{4}$
Simplify.

a) $63{v}^{10}\div 9{v}^{2}$

b) $-72{u}^{12}\div 12{u}^{4}$

c) $\frac{54{x}^{9}{y}^{15}}{-18{x}^{6}{y}^{3}}$

d) $\frac{20{m}^{8}{n}^{9}}{30{m}^{5}{n}^{4}}$

e) $\frac{45{x}^{8}{y}^{9}}{-60{x}^{5}{y}^{6}}$

f) $\frac{65{a}^{10}{b}^{8}{c}^{8}}{42{a}^{7}{b}^{6}{c}^{5}}$

g) $\frac{\left(-18{p}^{4}{q}^{7}\right)\left(-6{p}^{8}{q}^{8}\right)}{-36{p}^{7}{q}^{10}}$

h) $\frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}$
Simplify.

a) ${\left(\frac{3{m}^{5}}{5n}\right)}^{3}$

b) ${\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}$

c) ${\left(\frac{{j}^{2}{j}^{5}}{{j}^{4}}\right)}^{3}$

d) $\frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}$

e) $\frac{{\left(-2{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}$

f) $\frac{{\left(-10{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}$
When Paula simplified $-{3}^{0}$ and ${\left(-3\right)}^{0}$ , she got the same answer. Explain how using the Order of Operations correctly gives different answers.

Answers

1.

a) $72{u}^{7}$	b) $36{a}^{5}{b}^{7}$	c) $8{r}^{2}{s}^{5}$	d) $\dfrac{1}{2}{x}^{3}{y}^{3}$
e) $1024{a}^{10}$	f) $25000{p}^{24}$	g) ${x}^{18}{y}^{18}$	h) $144{m}^{8}{n}^{22}$

2.

a) $7{v}^{8}$	b) $-6{u}^{8}$	c) $-3{x}^{3}{y}^{12}$	d) $\frac{2{m}^{3}{n}^{5}}{3}$
e) $\frac{-3{x}^{3}{y}^{3}}{4}$	f) $\frac{65{a}^{3}{b}^{2}{c}^{3}}{42}$	g) $-3{p}^{5}{q}^{5}}$	h) $\frac{5{v}^{4}}{{u}^{2}}$