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lec 1

  • Common denominators and adding fractions

    • When adding fractions with different denominators, you need a common denominator so you can add the numerators: if you have fractions with denominators like 6 and 8, a common multiple is 24. The idea is: multiply the numerator and denominator of each fraction by the same factor so the denominator becomes the common denominator.

    • Example path to 24: to change a denominator from 6 to 24, multiply by 4. This means you multiply the numerator by 4 as well. So if the original numerator was 1, it becomes 1 × 4 = 4, and the denominator becomes 6 × 4 = 24.

    • Similarly, adjust the other fraction so its denominator becomes 24, then add the adjusted numerators over 24. This can yield a result like \frac{13}{24}.

    • Reduction after addition

    • After adding, you may get a fraction where the numerator and denominator share a common factor. To write the reduced (simplified) form, divide both numerator and denominator by their greatest common divisor (gcd). If there is no common factor other than 1 (e.g., gcd(13, 24) = 1), the fraction is already in reduced form.

    • Example: if you instead used a common denominator of 48, you might get \frac{26}{48}. Reducing by 2 gives \frac{13}{24}.

    • Key rule: always use a common denominator to add or subtract fractions; then reduce if possible.

  • Multiplying and dividing fractions

    • Multiplication rule

    • To multiply fractions, you multiply across (numerators with numerators and denominators with denominators): e.g., \frac{2}{3} \times \frac{3}{5} = \frac{2\cdot 3}{3\cdot 5} = \frac{6}{15}. You can simplify by canceling common factors before multiplying, e.g., cancel the 3's first: \frac{2}{\color{red}{3}} \times \frac{\color{red}{3}}{5} = \frac{2}{1} \times \frac{1}{5} = \frac{2}{5}.

    • Cancellation during multiplication

    • You can cancel common factors across the numerator and denominator before multiplying to simplify the result, as shown above.

    • Division of fractions (reciprocal)

    • Division by a fraction is equivalent to multiplying by its reciprocal: \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.

  • Exponents: zero, negative, and fractional exponents

    • Exponent zero

    • For any base a (except the caveat below), a^0 = 1. This is a standard convention.

    • The exception discussed: 0^0 is undefined in real numbers.

    • Negative exponents

    • A negative exponent means a reciprocal: a^{-n} = \frac{1}{a^n}. For example, a^{-1} = \frac{1}{a}.

    • Fractional exponents and roots

    • The expression a^{\frac{1}{n}} denotes the nth root of a, with the convention that for even n, the principal root is nonnegative.

    • Example: \sqrt{4} = \sqrt[2]{4} = 2 (principal root, positive).

    • Odd roots can be defined for negative inputs as well; e.g., \sqrt[3]{-8} = -2. ( cube root of -8 is -2 )

    • General form: a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}. Example: 8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4.

    • Specific example: 8^{\frac{2}{3}} = 4 as shown above; another form is \sqrt[3]{8^2} = \sqrt[3]{64} = 4.

  • Radical notation and rules

    • Radicals are roots; the notation (\sqrt[n]{\cdot}) denotes the nth root.

    • Fourth root example: \sqrt[4]{81} = 3, since 3^4 = 81.

    • Negative radicands and even roots

    • The even root of a negative number is undefined in the real numbers. For example, \sqrt{-25} is undefined.

    • Odd roots of negative numbers are defined (e.g., \sqrt[3]{-27} = -3).

  • Parity and simplifying nth roots with exponents

    • If you have the expression \sqrt[n]{a^n}:

    • If n is even, this equals |a| (the absolute value of a).

    • If n is odd, this equals a (the original number, preserving sign).

    • The concept of absolute value

    • Absolute value bars remove the sign: |a| is the nonnegative distance of a from zero on the real number line.

    • Examples: |2| = 2, |-2| = 2.

    • It represents the distance from zero, i.e., how far a number is from zero regardless of direction.

  • Absolute value and distance intuition (from the real number line)

    • The distance interpretation helps explain why (|a| = a) when a (\ge 0) and (|a| = -a) when a < 0.

    • This distance viewpoint is a practical way to reason about positivity and negativity in algebraic expressions.

  • Factorials (n!) and basic properties

    • Definition and convention

    • Factorial: n! = n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1 for positive integers n.

    • By convention, 0! = 1. This is defined to make many algebraic identities work smoothly (e.g., combinatorics and division with factorials).

    • A common mental model: factorials count the number of ways to arrange n distinct items (n! ways).

    • Quick arithmetic example

    • 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120. (Note: the transcript contained a verbal mix-up; 5! equals 120, not 24.)

    • Simplifying factorial expressions

    • An especially handy rule: when you have a ratio like \frac{n!}{(n-2)!} you can cancel the trailing factorial: \frac{n!}{(n-2)!} = n \cdot (n-1).

    • Example discussed: \frac{7!}{5!} = 7 \cdot 6 = 42. This follows from the cancellation:
      \frac{7!}{5!} = \frac{7 \cdot 6 \cdot 5!}{5!} = 7 \cdot 6 = 42.

    • Important note about 0! and definition motivation

    • The convention $0! = 1$ is chosen to keep formulas consistent (e.g., the number of ways to arrange zero items is one—there is exactly one empty arrangement).

  • Summary of key rules and connections

    • Exponent rules connect with roots: a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.

    • When adding or subtracting fractions, always find a common denominator first, then add/subtract the numerators, and finally reduce if possible.

    • When multiplying fractions, you may cancel common factors before multiplying to simplify the arithmetic.

    • When dividing by fractions, multiply by the reciprocal.

    • Zero and negative exponents convert to reciprocals; zero to the zero is undefined; negative bases under even roots are constrained by parity (even roots are undefined for negative radicands; odd roots are defined, preserving sign).

    • Absolute value measures distance from zero and converts negative inputs to positive outputs; it also helps interpret expressions like the nth root of a^n in terms of sign and magnitude.

    • Factorials grow quickly; 0! = 1 is a crucial convention; factorials enable compact evaluation of products like 7!/5!.

  • Preview of upcoming topics and plan

    • Friday’s class focus: Section 1.2 order of operations and Section 1.3 fundamental algebraic skills.

    • Expect a diagnostic at the end to assess understanding and identify areas needing review.

  • Quick self-check prompts (based on the transcript examples)

    • What is the common denominator for fractions with denominators 6 and 8? How do you scale each fraction to this common denominator? What is the reduced form after addition if the resulting numerator and denominator share no common factor?

    • How do you simplify the product \frac{2}{3} \cdot \frac{3}{5}? How does early cancellation affect the result?

    • What is the value of 8^{\frac{2}{3}} and why? Show both expressions: the fractional exponent form and the radical form.

    • Compute \sqrt[4]{81} and explain which root is used in the radical notation and why it’s defined that way.

    • Explain why \sqrt{-25} is undefined in reals, but \sqrt[3]{-27} = -3.

    • What is 0! and why is it defined as 1? Give an example where this convention makes a calculation easier, such as simplifying a factorial ratio.

    • State the general rule for the nth root of a^n depending on the parity of n.