Exponents and Exponent Rules — Comprehensive Lecture notes

Course logistics and grading

  • Grades are reported to the registrar and are calculated based on all objectives, including those from math 103A.
  • Scores on the displayed objectives are not revisited; the class structure mirrors 100 and a: three opportunities to demonstrate mastery: a quiz, an exam, and the final exam.
  • The final exam is individualized, as it was for math 103A, and will cover only the objectives not yet mastered or those you opt into at the end.
  • Any questions about grading or structure can be asked during class.
  • The instructor emphasizes openness to questions and the value of mistakes for learning.

Math Learning Center and class logistics

  • The Math Learning Center opens next Tuesday (the day after Labor Day).
  • It will have the same or similar locations as before; evening locations include Holden and Brody Acres, while Wells Hall is used during the day.
  • Online tutoring was offered previously but is no longer available due to low attendance.
  • If you have questions, you can ask in class or reach out to ULAs (Undergraduate Learning Assistants).

ULAs and classroom support

  • Two ULAs are assigned to this class (A B section): Jaden and Alexander.
  • They are Dow STEM Scholars and experienced in this class; they can help explain concepts and answer questions.
  • The instructor encourages asking questions and raising hands; ULAs can be interrupted if needed.
  • Mistakes are considered a natural part of learning; they help identify common misconceptions to address.

Exponents: motivation and notation

  • Exponents are a mathematical shorthand to indicate repeated multiplication.
  • A storytelling intuition: once we had numbers and addition/subtraction; to make multiplication easier, exponent notation was invented to indicate repeated multiplication without writing it out.
  • The “base” is the number or expression being multiplied, and the “exponent” tells how many times it is multiplied.
  • Example intuition: for a^n, the base is whatever is being multiplied, and the exponent n tells how many times to multiply that base by itself.

Reading and identifying bases and exponents

  • Example: read 2^5 = 2 × 2 × 2 × 2 × 2; the base is 2 and the exponent is 5.
  • For a^4 with base a, the base is a and the exponent is 4 (a is multiplied 4 times).
  • Case with multiple symbols: base can be a product inside parentheses. Example: (xy)^3 means the base is the product xy and it is multiplied 3 times: (xy)^3 = x^3 y^3.
  • When a negative sign is outside parentheses, it is not part of the base; when inside parentheses, it is part of the base.
    • Example: (-3)^3 means negative three multiplied by itself three times = -27.
    • If you write -3^3 (without parentheses), the interpretation is different: it means -(3^3) = -27.
  • Calculator pitfalls: typing -3^2 computes -(3^2) = -9; to compute (-3)^2 you must use parentheses: (-3)^2 = 9.
  • The negative sign outside the parentheses is not part of the base; if the negative is inside, it is part of the base.
  • For complex bases like 2x, the base is the entire expression if written as (2x)^n; if written as 2x^n (without parentheses), interpretation depends on order of operations and what is being raised to the power.

Exponent rules (three core shortcuts)

  • Product of powers (same base): if the bases are the same, you add exponents when multiplying
    • Rule: animesam=an+ma^n imes a^m = a^{n+m}
    • Example: if you multiply numbers with the same base, you add exponents.
  • Quotient of powers (same base): when dividing with the same base, subtract exponents
    • Rule: anam=an−m\frac{a^n}{a^m} = a^{n-m}
  • Power of a power: multiply the exponents when you raise a power to another power
    • Rule: (am)n=amn(a^m)^n = a^{mn}
  • Special case: distribution over a product inside the base
    • If the base is a product (ab) and the exponent n is a positive integer, then (ab)n=anbn(ab)^n = a^n b^n
    • Important caveat: this distribution is valid when what is inside the parentheses is a multiplication or division. It does not hold if the inside is addition or subtraction (e.g., $(x+y)^2
      eq x^2 + y^2$; FOIL gives the correct expansion: $(x+y)^2 = x^2 + 2xy + y^2$).
  • Fractions raised to a power
    • Rule: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Examples illustrating the rules

  • Same-base product: (23)(24)=23+4=27=128(2^3)(2^4) = 2^{3+4} = 2^7 = 128
  • Same-base quotient: 2723=27−3=24=16{2^7 \over 2^3} = 2^{7-3} = 2^4 = 16
  • Power of a power: (52)3=52⋅3=56=15625(5^2)^3 = 5^{2\cdot 3} = 5^6 = 15625
  • Fraction inside a base: (35)2=3252=925(\tfrac{3}{5})^2 = \tfrac{3^2}{5^2} = \tfrac{9}{25}
  • Distributing over a product inside the base: (xy)3=x3y3(xy)^3 = x^3 y^3
  • FOIL reminder for addition inside base: (x+y)^2 = x^2 + 2xy + y^2

Zero and negative exponents (patterns and interpretation)

  • Table pattern (positive exponents): each step to the right multiplies by the base; to the left, divide by the base.
  • Zero exponent: for any nonzero a, a0=1a^0 = 1
  • Negative exponents give reciprocals: a−n=1ana^{-n} = \frac{1}{a^n}
  • Move across a fraction by flipping the sign of the exponent: if you have a negative exponent, moving the term from numerator to denominator or vice versa changes the sign of the exponent.
    • Example: 1a−n=an\frac{1}{a^{-n}} = a^n and a−n=1ana^{-n} = \frac{1}{a^n}
  • A subtle point about expressions like $1/(2x)^{-3}$: you move the negative exponent across the fraction by changing its sign to obtain 1(2x)−3=(2x)3\frac{1}{(2x)^{-3}} = (2x)^3; equivalently, (2x)−3=1(2x)3(2x)^{-3} = \frac{1}{(2x)^3}, and moving it to the numerator would give $(2x)^3 rac{1}{1}$ when appropriate.
  • Important clarification about the exponent in the numerator vs. overall expression:
    • Negative exponent indicates how many times to multiply; you cannot separate the exponent into numerator/denominator if it is tied to the base as a whole (e.g., $x^n$ vs. $n^x$ are different kinds of operations).

Putting an exponential into the standard form y = a b^x

  • Why: the form y = a b^x is the standard exponential function form; it makes graph behavior obvious.
  • Example 1: Convert $2^{x+2}$ to the form $a b^x$.
    • Use product of powers in reverse: 2x+2=2x⋅22=4⋅2x2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x.
    • Therefore, $y = a b^x$ with $a = 4$ and $b = 2$.
  • Example 2: Convert $5^{2x}$ to the form $a b^x$.
    • Use power of a power: $5^{2x} = (5^2)^x = 25^x$.
    • This is already in the form $a b^x$ with $a = 1$, $b = 25$; if there is a separate constant factor, it can be absorbed into a as appropriate.
  • Example 3: If the expression is a constant over a power, e.g., 10025x\frac{100}{25^x}, rewrite as a product with a single base raised to x:
    • 10025x=100⋅25−x=100⋅(25−1)x=100(125)x\frac{100}{25^x} = 100 \cdot 25^{-x} = 100 \cdot (25^{-1})^{x} = 100 \left(\frac{1}{25}\right)^x.
    • Thus, $a = 100$ and b=125b = \frac{1}{25}.
  • Example 4: General strategy when you see a + or - in the exponent:
    • Try to isolate the x dependence in the exponent, often by using the product rule in reverse or the power rule to convert to a single base raised to x.
    • If the expression contains multiple bases with the same exponent, you can factor the common exponent into the base expression, producing a product of a constant and a term like b^x.
  • Takeaway: you can often express a function of the form y = something with x in the exponent as y = a b^x by:
    • factoring parts into a constant a, and
    • rewriting the remaining term as a base raised to x using the exponent rules (e.g., $(p^m)^n = p^{mn}$ or $p^{x+2} = p^x p^2$ reversed as needed).

Practice guidance and class workflow

  • After class, the instructor posts announcements in D2L describing the assignment for the next class.
  • Students are expected to practice between classes; focus on problems that do not involve fractional exponents and do not require solving equations.
  • If you finish early, you should practice further problems to reinforce understanding before the next class.
  • The rest of the objective will be finished on Thursday in class.

Practical implications and takeaways

  • Understanding exponent rules is foundational for higher math, including calculus and beyond.
  • Distinguishing when the negative sign is part of the base vs. outside the base is critical to avoid sign errors.
  • FOIL reminder: distribution of an exponent over an addition is not valid; instead, expand using distributive methods like FOIL for binomials, or rewrite using products of powers when appropriate.
  • When converting to the standard form y = a b^x, always look for opportunities to pull out constants and rewrite the remaining factor as a base raised to x.

Quick cheat sheet (highlights)

  • Key identities:
    • anam=an+ma^n a^m = a^{n+m}
    • anam=an−m\frac{a^n}{a^m} = a^{n-m}
    • (am)n=amn(a^m)^n = a^{mn}
    • (ab)n=anbn(ab)^n = a^n b^n if n is an integer and the base inside parentheses is a product
    • (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
    • a0=1(a≠0)a^0 = 1\quad (a \neq 0)
    • a−n=1ana^{-n} = \frac{1}{a^n}
    • (x+y)2=x2+2xy+y2(don’t distribute over addition inside a base)(x+y)^2 = x^2 + 2xy + y^2\quad\text{(don't distribute over addition inside a base)}
  • Example conversions:
    • 2x+2=2x⋅22=4⋅2x⇒y=4⋅2x2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x\Rightarrow y = 4\cdot 2^x
    • 52x=(52)x=25x⇒y=25x5^{2x} = (5^2)^x = 25^x\Rightarrow y = 25^x
    • 10025x=100⋅25−x=100(125)x⇒a=100,b=125\frac{100}{25^x} = 100 \cdot 25^{-x} = 100 \left(\frac{1}{25}\right)^x\Rightarrow a = 100, b = \frac{1}{25}

Assignment reminders (Thursday)

  • Use WebWork practice or the practice worksheets.
  • Avoid problems with fractional exponents or those that require solving equations for this session.
  • Practice problems to solidify whether you can identify equivalent expressions and apply exponent rules correctly.

If you have questions after class or while working through the material, please raise your hand or ask the ULAs for clarification. Your participation helps address common misconceptions for everyone.