Exponential Growth

• Case 1: +ve base < 1
  • E.g.: $(\frac{1}{2})^1, (\frac{1}{2})^2, (\frac{1}{2})^3, (\frac{1}{2})^4, (\frac{1}{2})^5, (\frac{1}{2})^6...$ or $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}...$
  • Higher the exponent smaller the number
• Case 2: -ve base < -1
  • E.g.: $(-3)^1, (-3)^2, (-3)^3, (-3)^4, (-3)^5$ or $-3, -9, -27, -81, -243$
  • Absolute value increases
  • +,- alternates
• Case 3: -ve base between -1 & 0
  • E.g.: $(-\frac{1}{2})^1, (-\frac{1}{2})^2, (-\frac{1}{2})^3, (-\frac{1}{2})^4, (-\frac{1}{2})^5, (-\frac{1}{2})^6...$ or $-\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, -`\frac{1}{32}, \frac{1}{64}...$
  • Absolute value decreases
  • +,- alternates

Exponent Properties

• Inverse law: $b^{-n}=\frac{1}{b^n}$ and $(\frac{p}{1})^{-n}=(\frac{q}{p})^n$
• Product law: $(ab)^n=a^nb^n$
• Quotient law: $(\frac{a}{b})^n=\frac{a^n}{b^n}$
• Factoring and Distributing: $P(a\pm b)=Pa \pm Pb$
• Warning: $(a \pm b)\neq a^n \pm b^n$
• Equating powers: If, $a^m=a^n$ then, $m=n$

Unit digit of Power

• Unit digit of any product is influenced by the unit digit of the two factors being multiplied. E.g.: $12\underline{3} \cdot 1\underline{6} = 3886\underline{8}$
• What is the unit digit of $57^{123}$?
  1. Considering only one's place, $7^1 =\underline{7}, 7^2=...\underline{9}, 7^3 = ...\underline{3}, 7^4=...\underline{1}, 7^5 =...\underline{7}$
  2. We restart pattern (1,7,9,3,1,7,9,3...) at multiples of 4. Therefore $7^{120}$ must be 1
  3. Therefore,$7^{120}$ has 1 in unit place, $7^{121}$ has 7, $7^{122}$ has 9 and $7^{123}$ has 3
  4. Therefore, the unit digit of $57^{123}$ is 3

Radicals

Radical Properties

• $\sqrt{PQ}=\sqrt{P}.\sqrt{Q}$
• $\sqrt{\frac{P}{Q}}=\frac{\sqrt{P}}{\sqrt{Q}}$
• $b^{\frac{1}{2}}=\sqrt{b}$
• $b^{\frac{1}{m}}=\sqrt[m]{b}$
• $b^{\frac{m}{n}}=(b^m)^{\frac{1}{n}}=(b^{\frac{1}{n}})^m$
• $\sqrt{k^2}=k$ (only if $k\geq0$)
  • E.g.: if $k=-4, \sqrt{-4^2}\neq -4$
• Common radical-to-decimals: $\sqrt{2}=1.4; \sqrt{3}=1.7; \sqrt{5}=2.2$

Rationalization

Always rationalize final value (eliminate radical from denominator)

• E.g: $\frac{1}{\sqrt{5}}=\frac{1}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{25}$
• Sometimes, use conjugate to rationalize the final value. E.g.: $a^2\cdot b^2=(a+b)(a-b)$

Extraneous roots in Radical equations

In a quadratic equation there are 3 possibilities with the final root(s):

• Both root work
• Only 1 root works. E.g.: $\sqrt{x+3}=x-3$ has these roots: {1, 6}, but only x = 6 works not x =1
• Both roots do not work (Even if roots exist, they don't work)

Preservation of Order of Inequality

• If $b>1$, $\sqrt{b}<b$
• If $0<b<1$, $\sqrt{b}>b$
• Roots preserve order of inequality
  • if $0<a<b<c$ order is preserved when $0<\sqrt[n]{a}<\sqrt[n]{b}<\sqrt[n]{c}$
  • E.g. $\sqrt[4]{50}$ is between what two positive integers?
    • 16 < 50 < 81 $\equiv$ $0<\sqrt[4]{16}<\sqrt[4]{50}<\sqrt[4]{81}$ So, $\sqrt[4]{50}$ is between 2 and 3