Recording-2025-08-21T14:56:06.030Z

Session overview

  • Class setup recap: If you don’t have an account, use your username and password for AppState to access homework on MyMathLab. We'll do worksheets in class, not before class. After taking notes, we’ll work on some worksheet problems.
  • Purpose of the opening review: Write down key concepts first, then we review them together later. This helps uncover what students thought they knew and what needs more study.
  • In-class activity structure:
    • Introductions with neighbors to compare answers and discuss definitions.
    • Instructor circulates to help students who are stuck.
    • Emphasis on collaborative study and verifying understanding.

Core concepts introduced (definitions and formulas)

  • Profit (in words)
    • Definition: money earned by a company after expenses; essentially, money left after costs are paid.
    • In words: profit = money made after expenses; the net earnings.
  • Profit (formula)
    • extProfit=extRevenue−extCostext{Profit} = ext{Revenue} - ext{Cost}
  • Revenue (definition and formula)
    • In words: money made from selling the product.
    • In formula: extRevenue=extPriceimesextQuantity=pimesxext{Revenue} = ext{Price} imes ext{Quantity} = p imes x where
    • pp = price of the item, xx = number of items sold.
  • Cost (definition and breakdown)
    • In words: money spent to produce a product or service.
    • In formula (two components): extCost=extVariablecost+extFixedcostext{Cost} = ext{Variable cost} + ext{Fixed cost}
    • Variable cost (depends on quantity): e.g. if producing one unit costs $cv$, total variable cost is c</em>vimesxc</em>v imes x.
    • Fixed cost (constant, not depending on production): e.g. rent, utilities; a constant FF that you pay even if no units are produced.
  • Practical example (revenue vs. cost)
    • Example (pricing and production costs): If you sell a T-shirt for p=20p = 20 and it costs cv=5c_v = 5 to make, then for quantity xx:
    • Revenue: extRevenue=pimesx=20xext{Revenue} = p imes x = 20x.
    • Variable cost: extVariablecost=cvimesx=5xext{Variable cost} = c_v imes x = 5x.
    • Total cost: extCost=extVariablecost+extFixedcost=5x+Fext{Cost} = ext{Variable cost} + ext{Fixed cost} = 5x + F where FF is the fixed cost (rent, utilities, etc.).
  • Core takeaway for course: these three concepts (profit, revenue, cost) will be used repeatedly throughout the course.

Polynomial arithmetic: addition, subtraction, and multiplication

  • Key rules (short recap from the class):
    • Like terms combine when adding/subtracting: e.g., x2+x2=2x2x^2 + x^2 = 2x^2.
    • Signs matter: adding negatives stays negative; subtracting a term changes its sign.
    • Negative times negative = positive; negative times positive = negative.
  • Example problems discussed:
    • Subtracting like terms: (−8x2)−(−7x2)=−8x2+7x2=−x2(-8x^2) - (-7x^2) = -8x^2 + 7x^2 = -x^2.
    • A pure negative term with a higher power: e.g., subtracting a term without an explicit sign (e.g., subtracting −15x4-15x^4) follows the same rule: combine like terms with correct signs.
    • When multiplying expressions with no plus/minus between terms (single-term cases), you can use the power rule for exponents: if you have (xa)b=xab(x^a)^b = x^{ab}, provided there is only one base term.
  • Product rule for exponents (single-term bases):
    • (xa)b=xab(x^a)^b = x^{ab} (valid only when there is one base term inside the parentheses; if there are plus/minus signs, use distributive/FOIL instead).
  • Distributive property (FOIL) for multiplying binomials:
    • For example, to multiply (Ax+B)(Cx+D)(Ax + B)(Cx + D), use FOIL: First, Outer, Inner, Last:
    • First: ACx2A C x^2
    • Outer: ADxA D x
    • Inner: BCxB C x
    • Last: BDB D
    • Then combine like terms to simplify.
  • Important practical rule when distributing with a monomial outside parentheses:
    • If you have something like −2x(x+2)-2x (x + 2), multiply each term inside first:
    • −2ximesx=−2x2-2x imes x = -2x^2
    • −2ximes2=−4x-2x imes 2 = -4x
    • Then combine terms if needed.
  • Reminder: you cannot add a letter to a number inside a single term; keep bases distinct (no x + 3 inside a product).

Exponents and powers: rules in detail

  • Power of a product and multi-term caution:
    • (ab)n=anbn(ab)^n = a^n b^n is valid when n is a positive integer, but only if the base is a single product term. If there are plus/minus terms inside the parentheses, do not apply this shortcut; distribute instead.
  • Zero power rule:
    • Any nonzero number to the zero power is 1: x0=1ext(forx<br/>eq0ext)x^0 = 1 ext{ (for } x <br /> eq 0 ext{)}
  • Negative exponents:
    • x^{-n} = rac{1}{x^n} ; move to the denominator to make the exponent positive.
  • Division of exponents (same base):
    • If you have racxmxnrac{x^m}{x^n} with the same base, then racxmxn=xm−nrac{x^m}{x^n} = x^{m-n}.
    • If the result exponent is negative, move the base to the opposite denominator accordingly.
  • Adding and subtracting exponents in division:
    • When you divide and have different bases, you cannot combine exponents across bases; you can only subtract exponents for the same base.
  • Negative exponent in numerator example:
    • If you have racx1x4=x−3rac{x^1}{x^4} = x^{-3} which is equivalent to rac1x3rac{1}{x^3} in positive exponent form.
  • Special cases with mixed signs and multiplication:
    • When multiplying like bases, add exponents.
    • When multiplying dissimilar bases, treat separately and then combine if possible.
  • Power of a sum is not straightforward: you must expand using FOIL/distributive property, not simply raise the sum to a power.

Radicals and fractional exponents

  • Root and fractional exponents connection:
    • A root is a fractional exponent: <br/>oot[n]x=x1/n<br /> oot[n]{x} = x^{1/n}
    • Example: extsquareroot(<br/>oot2x)=x1/2ext{square root } (<br /> oot{2}{x}) = x^{1/2}; more generally, <br/>oot[n]x=x1/n<br /> oot[n]{x} = x^{1/n}.
  • Examples discussed:
    • <br/>oot[3]y8=y8/3=y2+2/3=y2<br/>oot[3]y2<br /> oot[3]{y^8} = y^{8/3} = y^{2 + 2/3} = y^{2} <br /> oot[3]{y^2}
    • Fractional exponents: x^{ rac{m}{n}} =
      oot[n]{x^m}
  • Simplifying radicals (a practical approach):
    • When the exponent inside the root is larger than the root index, try to extract perfect powers from inside the root.
    • Example: <br/>oot[3]y8=y8/3=y2<br/>oot[3]y2<br /> oot[3]{y^8} = y^{8/3} = y^2 <br /> oot[3]{y^2} (extracts as many complete factors of 3 as possible and leaves remainder under the root).
    • Another example: <br/>oot[4]x16=x16/4=x4<br /> oot[4]{x^{16}} = x^{16/4} = x^{4} (fully simplified since 16 is a multiple of 4).
  • A practical method for rational exponents with mixed exponents:
    • For expressions like <br/>oot[n]xm<br /> oot[n]{x^m}, write m = qn + r with 0 ≤ r < n, then
    • <br/>oot[n]xm=xq<br/>oot[n]xr<br /> oot[n]{x^m} = x^{q} <br /> oot[n]{x^r}
  • When simplifying nested exponents or roots (root outside, inside powers), aim to extract as many whole powers as possible and keep a simplified radical for the remainder.
  • An example chain given: for a root with a large inside exponent, determine how many complete bases can come out and what remains inside to minimize the radical part.

Practice problems and worked walkthroughs (highlights from the session)

  • Problem: Evaluate the sign of (−3)9imes(−3)3(-3)^9 imes (-3)^3
    • Use exponent rules for the same base: add exponents: (−3)9+3=(−3)12(-3)^{9+3} = (-3)^{12}.
    • Since the exponent is even, the numeric value is positive: the sign is positive because a negative base raised to an even power yields a positive result.
  • Calculator tips (negative numbers and parentheses):