The following laws hold for a,b>0 and real indices. In root notation, n is a positive integer. Same base am an=am+n aman=am-n (am)n=amn **Examples** 5y+1 x 5y+4=52y+5 73y+27y-1=7(3y+2)-(y-1)=72y+3 (ex)3=e3x Same index an bn=(ab)n anbn=(ab)n **Examples** 5y x 2y=10y 15y5y=3y Zero, negative and fractional indices a0=1 a-n= 1an a mn= [n]am **Examples** 27(2)/(3)=([3]27)2=9 32(1)/(5)= [5]32=2 q-3= 1q3, 1x-6=x6 (35)-x=(53)x e0=1 Addition and subtraction **Skill** | **Working** Split shifted powers, then add like terms | 3y+2+3y+1 =(3y)(32)+(3y)(31) =9(3y)+3(3y) =12(3y) Split, then multiply powers with the same index | 5y+1 x 2y+2 =(5y)(51) x (2y)(22) =20(5y)(2y) =20(10y) Combine like terms; index laws do not combine a sum or difference into a single power.

The following laws hold for \(a,b>0\) and real indices. In root notation, \(n\) is a positive integer.

\(a^m a^n=a^{m+n}\qquad\dfrac{a^m}{a^n}=a^{m-n}\qquad(a^m)^n=a^{mn}\)

\(\dfrac{7^{3y+2}}{7^{y-1}}=7^{(3y+2)-(y-1)}=7^{2y+3}\)

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

\(a^0=1\qquad a^{-n}=\dfrac1{a^n}\qquad a^{\tfrac{m}{n}}=\sqrt[n]{a^m}\)

\(27^{\frac{2}{3}}=\left(\sqrt[3]{27}\right)^{2}=9\)

\(q^{-3}=\dfrac{1}{q^3},\quad \dfrac{1}{x^{-6}}=x^6\)

\(\left(\dfrac{3}{5}\right)^{-x}=\left(\dfrac{5}{3}\right)^{x}\)

Split shifted powers, then add like terms

\(\begin{aligned} &3^{y+2}+3^{y+1} \\ &=(3^y)(3^2)+(3^y)(3^1) \\ &=9(3^y)+3(3^y) \\ &=12(3^y) \end{aligned}\)

Split, then multiply powers with the same index

\(\begin{aligned} &5^{y+1}\times 2^{y+2} \\ &=(5^y)(5^1)\times(2^y)(2^2) \\ &=20(5^y)(2^y) \\ &=20(10^y) \end{aligned}\)

Combine like terms; index laws do not combine a sum or difference into a single power.

Exponential and Logarithmic Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Laws of Indices

Laws of Indices