1.7: Properties of Exponents

Multiplication Properties

Recall from chapter 1 that \(5^{3}=5 \cdot 5 \cdot 5 .\) In the same way we have

\[x^{3}=x \cdot x \cdot x\nonumber\]

Example 5.1

Perform the given operations:

1. \(x^{3} \cdot x^{5}=\underbrace{x \cdot x \cdot x}_{=x^{3}} \cdot \underbrace{x \cdot x \cdot x \cdot x \cdot x}_{=x^{5}}=x^{8}\)
2. \(\left(x^{3}\right)^{4}=x^{3} \cdot x^{3} \cdot x^{3} \cdot x^{3}=\underbrace{x \cdot x \cdot x} \cdot x \cdot x \cdot x \cdot \cdot \underbrace{x \cdot x \cdot x} \cdot x \cdot x \cdot x=x^{12}\)
3. \((x \cdot y)^{4}=\underbrace{x \cdot y} \cdot \underbrace{x \cdot y} \cdot \underbrace{x \cdot y} \cdot \underbrace{x \cdot y}=x^{4} y^{4}\)

We can summarize these examples into the following useful rules:

Multiplication and Exponentiation Rules

For any integers \(n\) and \(m\)

1. \(x^{n} x^{m}=x^{n+m}\)
2. \((x^{n})^{m}=x^{nm}\)
3. \((x y)^{n}=x^{n} y^{n}\)

Example 5.2

Perform the given operation using the multiplication properties of exponents and write your answer in simplest form:

1. \(b^{2} \cdot b^{3}=b^{2+3}=b^{5}\) (recall the meaning of exponents)
2. \(x^{8} x^{7}=x^{8+7}=x^{15}\) (note that juxtaposition indicates multiplication)
3. \(a^{8} a^{9} a^{14}=a^{8+9+14}=a^{31}\)
4. \(4 x^{4} \cdot 7 x^{6}=(4 \cdot 7)\left(x^{4} \cdot x^{6}\right)=28 x^{10}\)
5. \(5 x y^{3} \cdot 6 y=(5 \cdot 6)(x)\left(y^{3} \cdot y\right)=30 x y^{4}\)
6. \(\left(5 x^{4} y^{2}\right)\left(2 x^{7} y^{3}\right)=10 x^{4+7} y^{2+3}=10 x^{11} y^{5}\)
7. \((2 x)^{3}=2^{3} x^{3}=8 x^{3}\)
8. \(\left(-4 a^{2} b^{5}\right)^{3}=(-4)^{3} a^{2 \cdot 3} b^{5 \cdot 3}=-64 a^{6} b^{15}\)
9. \(\begin{align*}\left(-5 r^{3} s\right)^{2} \cdot\left(2 r^{4} s^{3}\right)^{3} \cdot\left(-r^{2} s^{2}\right) &=(-5)^{2} r^{3 \cdot 2} s^{2} \cdot 2^{3} r^{4 \cdot 3} s^{3 \cdot 3} \cdot(-1) r^{2} s^{2} \\&=25 r^{6} s^{2} \cdot 8 r^{12} s^{9} \cdot(-1) r^{2} s^{2} \\&=-200 r^{20} s^{13}\end{align*}\)

Division Properties

Example 5.3

1. \(\dfrac{x^{5}}{x^{3}}=\dfrac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x}=\dfrac{x \cdot x}{1}=\dfrac{x^{2}}{1}=x^{2}\)
2. \(\dfrac{x^{3}}{x^{5}}=\dfrac{x \cdot x \cdot x}{x \cdot x \cdot x \cdot x \cdot x}=\dfrac{1}{x \cdot x}=\dfrac{1}{x^{2}}\)
3. \(\left(\dfrac{x}{y}\right)^{5}=\dfrac{x}{y} \cdot \dfrac{x}{y} \cdot \dfrac{x}{y} \cdot \dfrac{x}{y} \cdot \dfrac{x}{y}=\dfrac{x \cdot x \cdot x \cdot x \cdot x}{y \cdot y \cdot y \cdot y \cdot y}=\dfrac{x^{5}}{y^{5}}\)

We can summarize these examples into the following useful rules:

Division Rules

For any integers \(n\) and \(m\)

1. \(\dfrac{x^{n}}{x^{m}}=x^{n-m}\)
2. \(\left(\dfrac{x}{y}\right)^{n}=\dfrac{x^{n}}{y^{n}}\)

Example 5.4

Perform the given operation using the division properties of exponents and state your answer in simplest form:

1. \(\dfrac{b^{8}}{b^{7}}=b^{8-7}=b^{8+(-7)}=b^{1}=b\)
2. \(\dfrac{x^{12} y^{2}}{x^{8} y}=x^{12-8} y^{2-1}=x^{12+(-8)} y^{2+(-1)}=x^{4} y^{1}=x^{4} y\)
3. \(\dfrac{8 x^{6} y^{2}}{6 x^{5} y^{7}}=\dfrac{4 x^{6-5}}{3 y^{7-2}}=\dfrac{4 x^{1}}{3 y^{5}}=\dfrac{4 x}{3 y^{5}}\)
4. \(\dfrac{\left(m^{2}\right)^{3}\left(n^{4}\right)^{5}}{\left(m^{3}\right)^{3}}=\dfrac{m^{6} n^{20}}{m^{9}}=\dfrac{n^{20}}{m^{3}}\)
5. \(\left(\dfrac{3 a^{2} b^{4}}{9 c^{3}}\right)^{2}=\left(\dfrac{a^{2} b^{4}}{3 c^{3}}\right)^{2}=\dfrac{a^{2 \cdot 2} b^{4 \cdot 2}}{3^{2} c^{3 \cdot 2}}=\dfrac{a^{4} b^{8}}{9 c^{6}}\)Another way to simplify this correctly is this: \(\left(\dfrac{3 a^{2} b^{4}}{9 c^{3}}\right)^{2}=\dfrac{3^{2} a^{2 \cdot 2} b^{4 \cdot 2}}{9^{2} c^{3 \cdot 2}}=\dfrac{9 a^{4} b^{8}}{81 c^{6}}=\dfrac{a^{4} b^{8}}{9 c^{6}}\)

Zero Exponent

Recall from chapter 1 that \((-7)^{0}=1\) and \(8^{0}=1 .\) In the same way we have \(x^{0}=1\)

Zero Exponent

\(a^{0}=1 \nonumber\)

Example 5.5

Evaluate

1. \(15^{0}=1\)
2. \((-15)^{0}=1\)
3. \(-15^{0}=-1\)
4. \((15 x)^{0}=1\)
5. \(15 x^{0}=15 \cdot 1=15\)

Negative Exponents

Negative Exponents

For any integer \(n\)

1. \(a^{-n}=\dfrac{1}{a^{n}}\)
2. \(\dfrac{1}{a^{-n}}=a^{n}\)
3. \(\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}\)

Note 5.6. The negative exponent rules can be used to switch terms from numerator to denominator or vice versa, and is useful to write expressions using positive exponents only.

Example 5.7

1. \(\dfrac{x^{-3} y^{2}}{z^{4}}=\dfrac{y^{2}}{x^{3} z^{4}}\)
2. \(\dfrac{x^{4}}{y^{-5} z^{2}}=\dfrac{x^{4} y^{5}}{z^{2}}\)

Example 5.8

Perform the given operation and write your answer using positive exponents only

1. \(\dfrac{x^{4} y^{2}}{x^{3} y^{-3}}=x^{4-3} y^{2-(-3)}=x^{4+(-3)} y^{2+3}=x y^{5}\)
2. \(\left(x^{2} y\right) \cdot\left(x y^{-4}\right)=x^{2+1} y^{1+(-4)}=x^{3} y^{-3}=\dfrac{x^{3}}{y^{3}}\)

Exit Probelm

Simplify: \(\left(\dfrac{8 y}{3 x^{3}}\right)^{3}\)