6.1.1: Resources and Key Concepts

Math 375: Pre-Calculus · LibreTexts

0%

Contents

    Text

    Size
    100%
    Theme
    Typeface
    Line length
    Spacing

    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Definitions

    • Conjugate (of an expression involving square roots): The conjugate of \(a+b\) is \(a-b\). Likewise, the conjugate of \(a-b\) is \(a+b\). This definition is extended from complex conjugates to expressions involving square roots (e.g., the conjugate of \(2-\sqrt{3}\) is \(2+\sqrt{3}\)).
    • Radical Functions: Functions involving roots (e.g., square root, cube root).

    Theorems

    • Roots of Powers: For any natural number \(n \ge 2\), \(\sqrt[n]{a^n} = \begin{cases} a, & \text{if } n \text{ is odd} \\ |a|, & \text{if } n \text{ is even} \end{cases}\).
    • Conjugate Multiplication: Given algebraic expressions \(a\) and \(b\), the product of \(a+b\) with its conjugate, \(a-b\), is always \(a^2 - b^2\). (This is essentially the difference of squares factorization.)

    Common Mistakes

    • Forgetting Absolute Value with Even Roots: When simplifying an even-indexed root of a variable raised to an even power, forgetting to use absolute value signs. For example, \(\sqrt{x^6} = |x^3|\), not just \(x^3\), unless it's known that \(x \ge 0\).
    • Incorrectly Applying Absolute Value with Odd Roots: Using absolute value signs when simplifying odd-indexed roots. For odd roots, \(\sqrt[n]{a^n} = a\).
    • Errors in Rationalizing:
      • Multiplying only the numerator or only the denominator by the conjugate, instead of both.
      • Incorrectly multiplying out the conjugate pair (e.g., not getting \(a^2-b^2\)).
      • Distributing unnecessarily in the non-conjugate part, which might obscure common factors that could cancel.
    • Errors in Simplifying Compound Fractions (in Difference Quotients):
      • Not multiplying by the correct LCD of all minor fractions.
      • Incorrectly distributing or canceling terms when simplifying the compound fraction.
    • Not Simplifying Radicals Completely: Leaving perfect \(n^{th}\) powers inside an \(n^{th}\) root (e.g., \(\sqrt{16n^2}\) should be simplified to \(4|n|\)).

    6.1.1: Resources and Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.