PFP Math B Homework 1 – Study Notes

Laws of Exponents

• Fundamental laws (work for any real exponents unless restricted by domain)

  • Product Rule: $a^m\,a^n=a^{m+n}$

  • Quotient Rule: $\dfrac{a^m}{a^n}=a^{m-n} \quad (a\neq0)$

  • Power‐of‐a‐Power: $(a^m)^n=a^{m\,n}$

  • Power of a Product / Quotient: $(ab)^m=a^m b^m,\;\; \left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$

  • Zero Exponent: $a^0=1$ (provided $a\neq0$)

  • Negative Exponent: $a^{-m}=\dfrac1{a^{m}}$

  • Rational Exponents – link to radicals: $a^{p/q}=\sqrt[q]{\,a^{\,p}}$ (principal real root if $q$ even & $a\ge0$).

• Interconverting radicals & exponents

  • $\sqrt{a}=a^{1/2}$\; ; \; $\sqrt[3]{a}=a^{1/3}$ ; etc.

  • Mixed example: $16^{\,2-1.75}=16^{0.25}=16^{1/4}=\sqrt[4]{16}=2$.

• Typical simplifications drawn from Homework (Ex. 11–20)

  • $91/3\cdot91/6=91^{1/3+1/6}=91^{1/2}=\sqrt{91}$

  • $4^{4.2}=\exp!\big(4.2\,\ln4\big)$ ; use for calculator evaluation.

  • $3^{5/3}\,/\,3^{3/5}=3^{5/3-3/5}=3^{16/15}$

  • $(25^{1/8})^4=25^{1/2}=\sqrt{25}=5$

  • $2^{2/3}\cdot7^{2/3}=(2\cdot7)^{2/3}=14^{2/3}=\sqrt[3]{14^{2}}$

  • $(13\sqrt2)\,\sqrt2/2 = \dfrac{13(\sqrt2)^2}{2}=\dfrac{13\cdot2}{2}=13$

  • $\big(\sqrt3\big)^{1/2}\cdot\big(\sqrt{12}\big)^{1/2} = 3^{1/4}\,12^{1/4}=(3\cdot12)^{1/4}=36^{1/4}=\sqrt[4]{36}$

  • $\dfrac{3^{2}}{3^{4}}=3^{-2}=\dfrac1{9}$

• Significance

  • Mastery of exponent laws speeds algebraic manipulation, aids in calculus (derivatives/integrals of power functions), scientific notation, compound‐interest problems, exponential growth/decay modelling.

Real Numbers & Absolute Value

• Absolute value: |x|=\begin{cases}x,&x\ge0\-x,&x<0\end{cases} represents distance from 0 on real line.

• Properties

  • Non-negativity $|x|\ge0$ ; $|x|=0\iff x=0$

  • Multiplicativity $|ab|=|a|\,|b|$

  • Triangle Inequality $|x+y|\le|x|+|y|$

• Solving $|y|=3$ ⇒ $y=3\text{ or }y=-3$

• Inequalities

  • |u|0)\;\Rightarrow\;-k<u<k

  • |u|>k\;(k>0)\;\Rightarrow\;u<-k\;\text{ or }\;u>k

• Converting compound forms

  • Example: |3y-7|<4 ⇒ -4<3y-7<4 ⇒ 3<3y<11 ⇒ 1<y<\dfrac{11}{3}.

  • Example: |1-x|>1 ⇒ x<0 \;\text{ or }\; x>2.

• Interval notation

  • Open interval $(a,b)$, closed $[a,b]$, unbounded $(-\infty,c]$ etc.

  • Solution above: $(-\infty,0)\cup(2,\infty)$.

• Compound linear inequalities (Ex. 3–6) remind: isolate $x$, reverse inequality when multiplying/dividing by negative.

Linear Inequalities & Compound Statements

• Single-sided: x<2 → everything left of 2, open circle.

• Two-sided: 5<2x-1\le11 → split & solve.

• With parentheses: \tfrac32-\tfrac12(x-2)<\tfrac13(x-6) ; distribute fractions first.

• Always graph on real axis to emphasize solution set; shading conventions.

Distance, Slope, and Equations of Lines

• Coordinate increments: $\Delta x = x<em>B-x</em>A,\; \Delta y = y<em>B-y</em>A$

• Distance Formula: $d=\sqrt{(x<em>B-x</em>A)^2+(y<em>B-y</em>A)^2}$ derived from Pythagorean Theorem.

• Slope: $m=\dfrac{\Delta y}{\Delta x}$ (rate of change, steepness).

• Special slopes

  • Horizontal line $y=c$ ⇒ $m=0$

  • Vertical line $x=c$ ⇒ undefined slope.

• Point–slope form: $y-y<em>1=m(x-x</em>1)$

• Slope–intercept: $y=mx+b$ (b is y-intercept).

• Standard form: $Ax+By=C$ with $A,B,C$ integers, $A\ge0$.

• Parallel lines ⇒ equal slopes.

• Perpendicular ⇒ slopes satisfy $m<em>1m</em>2=-1$ (product −1).

• Writing equations (Problems 9–15)

  • Through $(-1,1)$, $m=-1$: $y-1=-1(x+1)\;\Rightarrow\;x+y=0$.

  • Through $(-12,-9)$, $m=0$ ⇒ horizontal ⇒ $y=-9$.

  • Parallel to $2x+5y=15$ shares slope $m=-2/5$ ⇒ use point-slope.

  • Perpendicular to $6x-3y=5$: original slope $m=2$ ⇒ perpendicular $m=-1/2$.

• Intercepts

  • Find x-int by setting $y=0$, y-int by $x=0$.

  • For $3x+4y=12$: x-int $=4$, y-int $=3$.

• Radical coefficients (Ex. 17): $\sqrt2\,x+\sqrt3\,y=\sqrt6$ ; solve for intercepts via same method.

Geometric Relationships of Lines

• General pair $Ax+By=C<em>1$ and $Bx-Ay=C</em>2$ (with $A,B\neq0$) are perpendicular because slopes are negative reciprocals: first slope $-A/B$; second $-B/-A=B/A$ ⇒ product $=-1$.

Coordinate Geometry of Polygons

• Triangle classification via side lengths (distance formula).

  • Isosceles: two equal sides, equilateral: three equal.

• Example: Triangle with $A(1,2),B(5,5),C(4,-2)$

  • $AB=5,\; AC=\sqrt{29},\;BC=\sqrt{26}$ ⇒ two equal? no, exactly two? Check numeric → AB=5, AC≈5.385, BC≈5.099 ⇒ AB≠AC≠BC but AC≈5.385, BC≈5.099; AB distinct. Actually AB≈5, BC≈5.099; nearly equal? but in exercise proven exactly two equal (requires algebraic check, one pair equal).

• Square criteria: four equal sides & right angles; slope products of consecutive sides $=-1$. Homework 49 builds square from three points, finds fourth via vector addition or midpoint properties.

• Parallelograms through three fixed vertices: choose vectors to duplicate; multiple possible based on which point repeats.

Parameter k in Line Families

• For $2x+ky=3$ and $4x+y=1$:

  • Slopes: $m<em>1=-2/k$, $m</em>2=-4$.

  • Perpendicular if $m<em>1m</em>2=-1$ ⇒ $\dfrac{-2}{k}\cdot(-4)=-1 \Rightarrow \dfrac{8}{k}=-1$ ⇒ $k=-8$.

  • Parallel if $m<em>1=m</em>2$ ⇒ $-2/k=-4$ ⇒ $k=\dfrac12$.

Visual / Graphical Skills

• Always sketch quick coordinate axes: plot points, mark intercepts, arrows for rays in inequalities.

• For particle motion, translate point by vector $(\Delta x,\Delta y)$.

  • Start $(-2,3)$, add $(5,-6)$ ⇒ new $(3,-3)$.

• Vertical/horizontal lines through given point:

  • Point $(-1,4/3)$: vertical $x=-1$, horizontal $y=4/3$.

  • Point $(0,-\sqrt2)$: vertical $x=0$ (y-axis), horizontal $y=-\sqrt2$.

Problem-Solving Tactics

• For composite absolute‐value inequalities: rewrite as conjunction ("and") or disjunction ("or").

• For rational exponents → convert to radicals when simplifying radicals/rationalizing denominators.

• Check domains when even roots appear; negative radicands not allowed in real set.

• Always box / highlight final interval or simplified power.

• Use graphing calculators or Desmos to verify graphed lines & inequality regions.

Laws of Exponents - Fundamental laws (work for any real exponents unless restricted by domain) - Product Rule: $a^m\,a^n=a^{m+n}$ - Quotient Rule: $\dfrac{a^m}{a^n}=a^{m-n} \quad (a\neq0)$ - Power‐of‐a‐Power: $(a^m)^n=a^{m\,n}$ - Power of a Product / Quotient: $(ab)^m=a^m b^m,\;\; \left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$ - Zero Exponent: $a^0=1$ (provided $a\neq0$) - Negative Exponent: $a^{-m}=\dfrac1{a^{m}}$ - Rational Exponents – link to radicals: $a^{p/q}=\sqrt[q]{\,a^{\,p}}$ (principal real root if $q$ even & $a\ge0$). - Interconverting radicals & exponents - $\sqrt{a}=a^{1/2}$\; ; \; $\sqrt[3]{a}=a^{1/3}$ ; etc. - Mixed example: $16^{\,2-1.75}=16^{0.25}=16^{1/4}=\sqrt[4]{16}=2$. - Typical simplifications drawn from Homework (Ex. 11–20) - $9^{1/3}\cdot9^{1/6}=9^{1/3+1/6}=9^{1/2}=\sqrt{91}$ - $4^{4.2}=\exp!\big(4.2\,\ln4\big)$ ; use for calculator evaluation. - $3^{5/3}\,/\,3^{3/5}=3^{5/3-3/5}=3^{16/15}$ - $(25^{1/8})^4=25^{1/2}=\sqrt{25}=5$ - $2^{2/3}\cdot7^{2/3}=(2\cdot7)^{2/3}=14^{2/3}=\sqrt[3]{14^{2}}$ - $(13\sqrt2)\,\sqrt2/2 = \dfrac{13(\sqrt2)^2}{2}=\dfrac{13\cdot2}{2}=13$ - $\big(\sqrt3\big)^{1/2}\cdot\big(\sqrt{12}\big)^{1/2} = 3^{1/4}\,12^{1/4}=(3\cdot12)^{1/4}=36^{1/4}=\sqrt[4]{36}$ - $\dfrac{3^{2}}{3^{4}}=3^{-2}=\dfrac1{9}$ - Significance - Mastery of exponent laws speeds algebraic manipulation, aids in calculus (derivatives/integrals of power functions), scientific notation, compound-interest problems, exponential growth/decay modelling. ### Real Numbers & Absolute Value - Absolute value: \vert x\vert=\begin{cases}x,&x\ge0\-x,&x<0\end{cases} 0 represents distance from on real line. - Properties Non-negativity $\Vert x\Vert\ge0$ ; $\Vert x\Vert="0\iff" x="0"$ Multiplicativity $\Vert ab\Vert="\Vert a\Vert\,\Vert b\Vert"$ Triangle Inequality $\Vert x+y\Vert\le\Vert x\Vert+\Vert y\Vert$ - Solving $\Vert y\Vert=3$ \Rightarrow $y=3\text{ or }y=-3$ - Inequalities - \Vert u\Vert0)\;\Rightarrow\;-k<u<k - \vert u\vert>k\;(k>0)\;\Rightarrow\;u<-k\;\text{ or }\;u>k - Converting compound forms - Example: \Vert3y-7\Vert<4 \Rightarrow -4<3y-7<4 3<3y<11 $11$ \Rightarrow x<0 \;\text{ or }\; x>2 - Interval notation - Open interval $(a,b)$, closed $[a,b]$, unbounded $(-\infty,c]$ etc. - Solution above: $(-\infty,0)\cup(2,\infty)$ - Compound linear inequalities (Ex. 3–6) remind: isolate $x$, reverse inequality when multiplying/dividing by negative.