The Timpson Lyceum · College Algebra

Lecture 9 (College Algebra)

Overview

This lecture wraps up the number-system material and moves into the foundations of equation solving. It recaps complex numbers — the imaginary unit $i$, powers of $i$, and the arithmetic of complex numbers — then develops the general theory of $n$th roots, including how many roots a number has and which one we single out as the principal root. From there it builds rational exponents, the rules for simplifying radicals, and the technique of rationalizing a denominator (including the conjugate trick). It closes by defining equations, open equations, and solutions, and by warning which algebraic moves can secretly add or destroy solutions.

Key Concepts

The imaginary unit and complex numbers

• Definition of $i$: $i = \sqrt{-1}$, so $i^2 = -1$.
• Powers of $i$ cycle with period 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. To evaluate a high power, reduce the exponent modulo 4. For example $i^{100} = (i^4)^{25} = 1$ and $i^{1002} = i^{1000}\cdot i^2 = 1\cdot(-1) = -1$.
• Complex number: for real numbers $a$ and $b$, the expression $a + bi$ is a complex number; $a$ is the real part and $b$ the imaginary part.
• Number-system chain: every real number $a$ equals $a + 0i$, so $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$. Being a subset does not imply being "smaller" — e.g. $\mathbb{Z}$ and $\mathbb{Q}$ have the same cardinality.

Complex arithmetic

• Addition: $(a+bi) + (c+di) = (a+c) + (b+d)i$ — add real parts, add imaginary parts.
• Multiplication: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$. The minus sign appears because the $bi\cdot di = bd\,i^2$ term contributes $-bd$ once $i^2$ is replaced by $-1$.

Square roots of negative numbers

• For $p > 0$, $\sqrt{-p} = i\sqrt{p}$, and both $i\sqrt{p}$ and $-i\sqrt{p}$ square to $-p$, so every negative number has two square roots. For example the square roots of $-5$ are $\pm i\sqrt{5}$, and the square roots of $-4$ are $\pm 2i$.
• Caution: the rule $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ is not reliable when the radicands are negative; carry the $i$ explicitly instead.

$n$th roots and the principal root

• Definition: $r$ is an $n$th root of $a$ provided $r^n = a$. A number has $n$ distinct $n$th roots (e.g. $8$ has three cube roots: $2$, $-1+\sqrt{3}\,i$, and $-1-\sqrt{3}\,i$).
• Principal $n$th root $\sqrt[n]{a}$ — the single root the radical symbol denotes:
  • If $a > 0$: the unique positive real root.
  • If $a = 0$: it is $0$.
  • If $a < 0$ and $n$ is odd: the (unique) negative real root, e.g. $\sqrt[3]{-8} = -2$.
  • If $a < 0$ and $n$ is even: only complex roots exist, so no real principal root is defined. By standing agreement this course only writes $\sqrt[n]{a}$ with even $n$ when $a \ge 0$.
• When no index is written, $\sqrt{\ \ }$ means the principal square root ($n = 2$).

Radical and exponent rules (with $a, b > 0$ where an even index is involved)

• $\big(\sqrt[n]{a}\big)^n = a$ and $\sqrt[n]{a^n} = a$.
• $\sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}$ and $\sqrt[n]{\tfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$ (with $b \ne 0$).
• Rational exponents: $\sqrt[n]{a^m} = a^{m/n} = \big(\sqrt[n]{a}\big)^m$. Choosing whether to take the root first or the power first can make the arithmetic far easier — e.g. $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$ is easier than $\sqrt[3]{8^2}$.
• Negative exponents: $a^p \cdot a^{-p} = a^0 = 1$ (for $a \ne 0$), which gives $a^{-p} = \dfrac{1}{a^p}$. To simplify, flip a factor across the fraction bar to make its exponent positive.

Rationalizing the denominator

• Goal: remove radicals from the denominator. For a single root, multiply top and bottom by the radical needed to complete a perfect power — e.g. multiply $\dfrac{\sqrt{3}}{\sqrt{7}}$ by $\dfrac{\sqrt{7}}{\sqrt{7}}$ to get $\dfrac{\sqrt{21}}{7}$.
• For a two-term denominator, multiply by its conjugate: the conjugate of $a\sqrt{b} + c\sqrt{d}$ is $a\sqrt{b} - c\sqrt{d}$. The cross terms cancel, leaving $a^2 b - c^2 d$ with no radicals.

Equations and solutions

• An equation is a statement that two quantities are equal (and is either true or false). An open equation contains at least one variable, so its truth depends on the variable's value.
• A solution is an assignment of values to the variables that makes the equation true; to solve an equation is to find the set of all its solutions.
• Operations that can change the solution set: multiplying or dividing both sides by an expression that can equal zero (like $x$) may introduce or lose solutions. Multiplying/dividing by a constant guaranteed nonzero (like $5$ or $30$) is always safe.

Worked Example

Simplify $\left(-\dfrac{1}{125}\right)^{-2/3}$.

The negative exponent is cleared by taking the reciprocal of the base, which flips the exponent to $+2/3$; then $125/(-1) = -125$. Evaluating as a cube root first keeps the numbers small: the cube root of $-125$ is $-5$, and squaring gives $25$.

Summary

• $i = \sqrt{-1}$, $i^2 = -1$, and powers of $i$ repeat every 4 — reduce the exponent mod 4 to evaluate.
• Complex numbers add componentwise and multiply by $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$; the real numbers sit inside the complex numbers.
• Every nonzero number has $n$ distinct $n$th roots; the radical $\sqrt[n]{a}$ denotes the single principal root, chosen as the positive real (or the negative real for odd $n$ and $a<0$).
• Rational exponents convert radicals to powers: $\sqrt[n]{a^m} = a^{m/n}$; take the root before the power when it simplifies the arithmetic, and use $a^{-p} = 1/a^p$ to clear negative exponents.
• Rationalize a denominator by multiplying by the matching radical, or by the conjugate for two-term denominators.
• To solve an open equation is to find all values making it true; multiplying or dividing both sides by an expression that could be zero can add or remove solutions, so only do so with a guaranteed-nonzero factor.