Laws of Positive Indices

Powers mean repeated multiplication:

an = a × a × ⋯ × a

(n factors of a)

Why \(a^{m} \times a^{n} = a^{m+n}\)

Example: a3 × a2

a3
a × a2
a × a × a
a × a × a · a2
a × a × a · a × a

count factors: 3 + 2 = 5

\(a^{\textcolor{#ffd54f}{5}} = a^{\textcolor{#ffd54f}{3+2}}\)
\(a^{m} \times a^{n} = a^{m+n}\)

Why \(a^{m} \div a^{n} = a^{m-n}\)

Example: a4 ÷ a2

a4 a × a × a × a
a2 a × a
=
\(a^{\textcolor{#ffd54f}{2}}\)

cancel matching factors on top and bottom

\(a^{\textcolor{#4fc3f7}{4}-\textcolor{#81c784}{2}} = a^{\textcolor{#ffd54f}{2}}\)
\(a^{m} \div a^{n} = a^{m-n}\quad(a\neq 0)\)

Negative indices

y3 ÷ y5

use the quotient law:

\(a^{m}\div a^{n}=a^{m-n}\)

y−2

from \(y^{3}\div y^{5}=y^{3-5}\)

y × y × y
y × y × y × y × y
=
1y2
y−2 = 1y2

same leftover as a reciprocal power

\(a^{-n}=\dfrac{1}{a^{n}}\quad(a\neq 0)\)

Why \((a^{m})^{n} = a^{mn}\)

Example: (a3)2

(a3)2
(a3) × (a3)

use the product law: \(a^{m}\times a^{n}=a^{m+n}\)

\(=a^{3}\times a^{3}\)
\(=a^{\textcolor{#ffd54f}{3+3}}\)
\(=a^{\textcolor{#ffd54f}{3\times 2}}=a^{\textcolor{#ffd54f}{6}}\)
\((a^{m})^{n}=a^{mn}\)

Why \((ab)^{n} = a^{n} b^{n}\)

Example: (ab)2

(ab)2
= (ab) × (ab)
= a × a · b × b

regroup like bases: a's together, b's together

\((ab)^{n}=a^{n}b^{n}\)

Why \(\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}\)

Example: ( a b ) 2

( ab )2
( ab ) × ( ab )
= a2 b2

same power on numerator and denominator

\(\left(\dfrac{a}{b}\right)^{n}=\dfrac{a^{n}}{b^{n}}\)

All Five Laws

\[ a^{m} \times a^{n} = a^{m+n} \]
\[ a^{m} \div a^{n} = a^{m-n} \]
\[ (a^{m})^{n} = a^{mn} \]
\[ (ab)^{n} = a^{n} b^{n} \]
\[ \left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}} \]