Unit 1 – The Complex Number System Review

Simplifying Radical Expressions

• Radical Definition & Standard Form:

  • An expression containing a radical symbol $\sqrt[n]{x}$ is in simplest radical form when:
    1. The radicand has no factors with an exponent greater than or equal to the index $n$
    2. The radicand contains no fractions
    3. No radicals appear in the denominator of a fraction

• Problem 1: Simplifying a Square Root

  • Expression: $\sqrt{24}$
  • Step-by-step solution:
    • Identify the largest perfect square factor of $24$: $24 = 4 \times 6$
    • Apply the product property of radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
    • $\sqrt{24} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}$

• Problem 2: Simplifying a Cube Root with a Negative Radicand

  • Expression: $\sqrt[3]{-162}$
  • Step-by-step solution:
    • Identify the largest perfect cube factor of $-162$: $-162 = -27 \times 6$
    • Apply the product property for cube roots: $\sqrt[3]{-27 \times 6} = \sqrt[3]{-27} \cdot \sqrt[3]{6}$
    • Since $(-3)^3 = -27$, $\sqrt[3]{-27} = -3$
    • $\sqrt[3]{-162} = -3\sqrt[3]{6}$

• Problem 3: Simplifying a Square Root with Variable Terms

  • Expression: $\sqrt{98x}$
  • Step-by-step solution:
    • Factor the coefficient into perfect square factors: $98 = 49 \times 2$
    • Separate the perfect square terms: $\sqrt{98x} = \sqrt{49 \cdot 2 \cdot x} = \sqrt{49} \cdot \sqrt{2x}$
    • Evaluate the square root: $\sqrt{49} = 7$
    • $\sqrt{98x} = 7\sqrt{2x}$

• Problem 4: Simplifying a Fourth Root with Variables

  • Expression: $\sqrt[4]{128n^8}$
  • Step-by-step solution:
    • Factor the numerical constant into fourth-power factors: $128 = 16 \times 8 = 2^4 \times 8$
    • Rewrite the variable exponent as a power of 4: $n^8 = (n^2)^4$
    • Apply the product property: $\sqrt[4]{128n^8} = \sqrt[4]{16 \cdot n^8 \cdot 8} = \sqrt[4]{16} \cdot \sqrt[4]{n^8} \cdot \sqrt[4]{8}$
    • Simplify each term: $\sqrt[4]{16} = 2$ and $\sqrt[4]{n^8} = n^2$
    • $\sqrt[4]{128n^8} = 2n^2\sqrt[4]{8}$

• Problem 5: Subtracting Like Radicals

  • Expression: $3\sqrt{6} - 4\sqrt{6}$
  • Step-by-step solution:
    • Radicals can be combined by combining their coefficients when they have the exact same index and radicand
    • Factor out the common radical term $\sqrt{6}$: $(3 - 4)\sqrt{6}$
    • $3\sqrt{6} - 4\sqrt{6} = -1\sqrt{6} = -\sqrt{6}$

• Problem 6: Adding Radicals after Simplifying

  • Expression: $2\sqrt{6} + 2\sqrt{24}$
  • Step-by-step solution:
    • Simplify $\sqrt{24}$ first: $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$
    • Substitute back into the expression: $2\sqrt{6} + 2(2\sqrt{6}) = 2\sqrt{6} + 4\sqrt{6}$
    • Combine like radicals: $(2 + 4)\sqrt{6} = 6\sqrt{6}$

• Problem 7: Multiplying Radical Expressions

  • Expression: $\sqrt{20x^2} \cdot \sqrt{20x}$
  • Step-by-step solution:
    • Combine under a single radical using the product rule: $\sqrt{20x^2 \cdot 20x} = \sqrt{400x^3}$
    • Factor into perfect square components: $400x^3 = 400 \cdot x^2 \cdot x$
    • Take square roots of perfect square components: $\sqrt{400} = 20$ and $\sqrt{x^2} = x$
    • $\sqrt{20x^2} \cdot \sqrt{20x} = 20x\sqrt{x}$

• Problem 8: Distributing a Radical Expression

  • Expression: $2\sqrt{5}(\sqrt{6} + 2)$
  • Step-by-step solution:
    • Apply the distributive property: $2\sqrt{5} \cdot \sqrt{6} + 2\sqrt{5} \cdot 2$
    • Multiply terms: $2\sqrt{5 \cdot 6} + (2 \cdot 2)\sqrt{5} = 2\sqrt{30} + 4\sqrt{5}$
    • Since $\sqrt{30}$ cannot be simplified further ($30 = 2 \times 3 \times 5$), the final answer is $2\sqrt{30} + 4\sqrt{5}$

• Problem 9: Binomial Radical Multiplication (FOIL)

  • Expression: $(5 - 4\sqrt{5})(-2 + \sqrt{5})$
  • Step-by-step solution:
    • First: $5 \cdot (-2) = -10$
    • Outer: $5 \cdot \sqrt{5} = 5\sqrt{5}$
    • Inner: $-4\sqrt{5} \cdot (-2) = 8\sqrt{5}$
    • Last: $-4\sqrt{5} \cdot \sqrt{5} = -4(5) = -20$
    • Sum all terms: $-10 + 5\sqrt{5} + 8\sqrt{5} - 20$
    • Combine real constants and like radical terms: $(-10 - 20) + (5 + 8)\sqrt{5} = -30 + 13\sqrt{5}$

• Problem 10: Product of Conjugates

  • Expression: $(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})$
  • Step-by-step solution:
    • Recognize the pattern as a difference of squares: $(a - b)(a + b) = a^2 - b^2$
    • Set $a = \sqrt{5}$ and $b = \sqrt{3}$
    • $(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2$

• Problem 11: Rationalizing a Monomial Square Root Denominator

  • Expression: $\frac{4}{\sqrt{5}}$
  • Step-by-step solution:
    • Multiply numerator and denominator by $\sqrt{5}$ to eliminate the radical in the denominator:
    • $\frac{4}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4\sqrt{5}}{5}$

• Problem 12: Rationalizing a Monomial Cube Root Denominator

  • Expression: $\frac{2}{\sqrt[3]{3}}$
  • Step-by-step solution:
    • To eliminate a cube root, the expression inside the radicand in the denominator must be a perfect cube ($3^3 = 27$
    • Multiply numerator and denominator by $\sqrt[3]{3^2} = \sqrt[3]{9}$:
    • $\frac{2}{\sqrt[3]{3}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}} = \frac{2\sqrt[3]{9}}{\sqrt[3]{27}} = \frac{2\sqrt[3]{9}}{3}$

• Problem 13: Rationalizing a Binomial Denominator Using Conjugates

  • Expression: $\frac{3}{2 + 2\sqrt{5}}$
  • Step-by-step solution:
    • Identify the conjugate of the denominator $2 + 2\sqrt{5}$, which is $2 - 2\sqrt{5}$
    • Multiply both numerator and denominator by the conjugate:
    • Numerator: $3(2 - 2\sqrt{5}) = 6 - 6\sqrt{5}$
    • Denominator: $(2 + 2\sqrt{5})(2 - 2\sqrt{5}) = 2^2 - (2\sqrt{5})^2 = 4 - (4 \cdot 5) = 4 - 20 = -16$
    • Simplify the fraction: $\frac{6 - 6\sqrt{5}}{-16} = \frac{-6 + 6\sqrt{5}}{16} = \frac{3\sqrt{5} - 3}{8}$

• Problem 14: Rationalizing a Complex Radical Binomial Quotient

  • Expression: $\frac{\sqrt{5} + 3}{4 - \sqrt{5}}$
  • Step-by-step solution:
    • Identify the conjugate of the denominator $4 - \sqrt{5}$, which is $4 + \sqrt{5}$
    • Multiply numerator and denominator by $4 + \sqrt{5}$:
    • Numerator (FOIL): $(\sqrt{5} + 3)(4 + \sqrt{5}) = 4\sqrt{5} + (\sqrt{5})^2 + 12 + 3\sqrt{5} = 4\sqrt{5} + 5 + 12 + 3\sqrt{5} = 17 + 7\sqrt{5}$
    • Denominator: $(4 - \sqrt{5})(4 + \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11$
    • $\frac{\sqrt{5} + 3}{4 - \sqrt{5}} = \frac{17 + 7\sqrt{5}}{11}$

Rational Exponents and Exponent Rules

• Fundamental Conversions & Rules:

  • Radical to Rational Exponent: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ and $(\sqrt[n]{a})^m = a^{\frac{m}{n}}$
  • Product Rule: $a^m \cdot a^n = a^{m+n}$
  • Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
  • Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$
  • Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$

• Problem 15: Converting Radical Expression to Exponential Form

  • Expression: $(\sqrt[4]{m})^3$
  • Step-by-step solution:
    • Apply the conversion rule $(\sqrt[n]{m})^p = m^{\frac{p}{n}}$
    • Here, index $n = 4$ and power $p = 3$
    • $(\sqrt[4]{m})^3 = m^{\frac{3}{4}}$

• Problem 16: Converting Radical Expression with Product to Exponential Form

  • Expression: $\sqrt[3]{(6x)^4}$
  • Step-by-step solution:
    • Apply the conversion rule $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ with base $a = 6x$, power $m = 4$, and index $n = 3$
    • $\sqrt[3]{(6x)^4} = (6x)^{\frac{4}{3}}$

• Problem 17: Converting Rational Exponent to Simplest Radical Form

  • Expression: $7^{\frac{1}{2}}$
  • Step-by-step solution:
    • Apply the rule $a^{\frac{1}{n}} = \sqrt[n]{a}$
    • $7^{\frac{1}{2}} = \sqrt{7}$

• Problem 18: Converting Rational Exponent to Simplest Radical Form

  • Expression: $(10n)^{\frac{3}{2}}$
  • Step-by-step solution:
    • Rewrite in radical form: $(10n)^{\frac{3}{2}} = \sqrt{(10n)^3}$
    • Expand the power inside: $\sqrt{1000n^3}$
    • Factor out perfect squares: $\sqrt{100 \cdot 10 \cdot n^2 \cdot n} = \sqrt{100n^2} \cdot \sqrt{10n}$
    • Simplify: $10n\sqrt{10n}$

• Problem 19: Simplifying Product of Monomials with Rational Exponents

  • Expression: $2m^2 \cdot m^{\frac{3}{2}}$
  • Step-by-step solution:
    • Use product rule for exponents: $m^2 \cdot m^{\frac{3}{2}} = m^{2 + \frac{3}{2}}$
    • Convert integer exponent to fraction with common denominator: $2 = \frac{4}{2}$
    • Add fractional exponents: $\frac{4}{2} + \frac{3}{2} = \frac{7}{2}$
    • Exponential form: $2m^{\frac{7}{2}}$
    • Convert to simplest radical form: $2\sqrt{m^7} = 2\sqrt{m^6 \cdot m} = 2m^3\sqrt{m}$

• Problem 20: Simplifying Quotient of Monomials with Rational Exponents

  • Expression: $\frac{2x^{\frac{7}{4}}}{4x^{\frac{4}{3}}}$
  • Step-by-step solution:
    • Divide numerical coefficients: $\frac{2}{4} = \frac{1}{2}$
    • Apply quotient rule to variable terms: $x^{\frac{7}{4} - \frac{4}{3}}$
    • Find common denominator for exponents ($\text{LCD} = 12$): $\frac{7}{4} = \frac{21}{12}$ and $\frac{4}{3} = \frac{16}{12}$
    • Subtract exponents: $\frac{21}{12} - \frac{16}{12} = \frac{5}{12}$
    • Exponential form: $\frac{1}{2}x^{\frac{5}{12}}$
    • Convert to simplest radical form: $\frac{\sqrt[12]{x^5}}{2}$

• Problem 21: Applying Power of a Power with Negative Rational Exponents

  • Expression: $\left(p^{\frac{3}{2}}\right)^{-2}$
  • Step-by-step solution:
    • Apply power rule $(a^m)^n = a^{m \cdot n}$: $p^{\frac{3}{2} \cdot (-2)} = p^{-3}$
    • Apply negative exponent rule $a^{-n} = \frac{1}{a^n}$: $\frac{1}{p^3}$

• Problem 22: Simplifying Multi-Step Rational Exponent Quotient

  • Expression: $\frac{x^{-\frac{1}{2}} \cdot x^{\frac{1}{2}}}{x^{-\frac{7}{4}}}$
  • Step-by-step solution:
    • Simplify numerator using product rule: $x^{-\frac{1}{2} + \frac{1}{2}} = x^0 = 1$
    • Rewrite expression: $\frac{1}{x^{-\frac{7}{4}}}$
    • Apply negative exponent rule: $x^{\frac{7}{4}}$
    • Convert to simplest radical form: $x^{\frac{7}{4}} = \sqrt[4]{x^7} = \sqrt[4]{x^4 \cdot x^3} = x\sqrt[4]{x^3}$

Complex Numbers and Operations Involving $i$

• Imaginary Unit Definition:

  • The imaginary unit $i$ is defined as $i = \sqrt{-1}$, where $i^2 = -1$
  • Standard form of a complex number: $a + bi$, where $a$ is the real part and $b$ is the imaginary part ($a, b \in \mathbb{R}$)

• Cyclic Nature of Powers of $i$:

  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
  • For any integer exponent $n$, divide $n$ by $4$ to find the remainder $r$ ($r \in \{0, 1, 2, 3\}$): $i^n = i^r$

• Problem 23: Adding Complex Numbers

  • Expression: $3 + (4 + 6i)$
  • Step-by-step solution:
    • Combine real parts: $3 + 4 = 7$
    • Keep imaginary part: $6i$
    • Result in standard form $a + bi$: $7 + 6i$

• Problem 24: Subtracting Imaginary Terms

  • Expression: $-8i - 7i$
  • Step-by-step solution:
    • Combine like terms: $(-8 - 7)i = -15i$

• Problem 25: Distributing an Imaginary Term

  • Expression: $4i(-2 - 8i)$
  • Step-by-step solution:
    • Distribute $4i$: $4i(-2) - 4i(8i) = -8i - 32i^2$
    • Substitute $i^2 = -1$: $-8i - 32(-1) = -8i + 32$
    • Write in standard form $a + bi$: $32 - 8i$

• Problem 26: Multiplying Complex Binomials

  • Expression: $(-2 - i)(4 + i)$
  • Step-by-step solution:
    • Apply FOIL method:
      • First: $-2 \cdot 4 = -8$
      • Outer: $-2 \cdot i = -2i$
      • Inner: $-i \cdot 4 = -4i$
      • Last: $-i \cdot i = -i^2$
    • Combine terms: $-8 - 2i - 4i - i^2 = -8 - 6i - (-1)$
    • Simplify real constants: $-8 + 1 - 6i = -7 - 6i$

• Problem 27: Evaluating High Powers of $i$ (Odd Exponent)

  • Expression: $i^{31}$
  • Step-by-step solution:
    • Divide exponent by 4: $31 \div 4 = 7$ with a remainder of $3$ ($31 = 4 \times 7 + 3$
    • Rewrite using exponent rules: $i^{31} = (i^4)^7 \cdot i^3$
    • Since $i^4 = 1$ and $i^3 = -i$: $(1)^7 \cdot (-i) = 1 \cdot (-i) = -i$

• Problem 28: Evaluating High Powers of $i$ (Even Exponent)

  • Expression: $i^{42}$
  • Step-by-step solution:
    • Divide exponent by 4: $42 \div 4 = 10$ with a remainder of $2$ ($42 = 4 \times 10 + 2$
    • Rewrite using exponent rules: $i^{42} = (i^4)^{10} \cdot i^2$
    • Since $i^4 = 1$ and $i^2 = -1$: $(1)^{10} \cdot (-1) = 1 \cdot (-1) = -1$

The Complex Number System Hierarchy and Closure Properties

• Classification of the Number System:

  • Complex Numbers ($\mathbb{C}$): The overarching set of all numbers expressed as $a + bi$, where $a, b \in \mathbb{R}$
  • Real Numbers ($\mathbb{R}$): Complex numbers where $b = 0$ ($a + 0i$), represented on a continuous number line
  • Imaginary Numbers: Complex numbers where $b \neq 0$ and $a = 0$ (pure imaginary) or $a \neq 0$ (non-real complex)
  • Rational Numbers ($\mathbb{Q}$): Real numbers that can be written as a quotient of two integers $\frac{a}{b}$ with $b \neq 0$; decimal expansions terminate or repeat
  • Irrational Numbers ($\mathbb{I}$): Real numbers that cannot be expressed as a fraction of integers; decimal expansions are non-terminating and non-repeating
  • Integers ($\mathbb{Z}$): Rational numbers with no fractional or decimal part: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
  • Whole Numbers ($\mathbb{W}$): Non-negative integers: $\{0, 1, 2, 3, 4, \dots\}$
  • Natural Numbers ($\mathbb{N}$): Counting numbers excluding zero: $\{1, 2, 3, 4, \dots\}$

• Problem 29: Sets of Numbers and Examples

  • Natural Numbers ($\mathbb{N}$): $1, 5, 23, 100$
  • Whole Numbers ($\mathbb{W}$): $0, 1, 7, 42$
  • Integers ($\mathbb{Z}$): $-12, -3, 0, 8, 15$
  • Rational Numbers ($\mathbb{Q}$): $\frac{1}{2}, -\frac{4}{5}, 0.75, 0.\bar{3}, 6$
  • Irrational Numbers ($\mathbb{I}$): $\sqrt{2}, \sqrt{7}, \pi, e$
  • Real Numbers ($\mathbb{R}$): $-5, 0, \frac{2}{3}, \sqrt{3}, \pi$
  • Imaginary Numbers: $3i, -8i, i\sqrt{5}$
  • Complex Numbers ($\mathbb{C}$): $2 + 3i, -5 - 4i, 7, -9i$

• Problem 30: Closure Property of Whole Numbers under Subtraction

  • Statement: "Any whole number subtracted by another whole number will always result in a whole number."
  • Truth Value: False
  • Explanation: Whole numbers are not closed under subtraction.
  • Counterexample:
    • Select whole numbers $3 \in \mathbb{W}$ and $5 \in \mathbb{W}$
    • Subtract: $3 - 5 = -2$
    • $-2$ is an integer ($-2 \in \mathbb{Z}$), but it is NOT a whole number ($-2 \notin \mathbb{W}$

• Problem 31: Closure Property of Rational Numbers under Multiplication

  • Statement: "Any rational number multiplied by another rational number will always result in a rational number."
  • Truth Value: True
  • Explanation: Rational numbers are closed under multiplication.
  • Proof / Demonstration:
    • Let $r_1 = \frac{a}{b}$ and $r_2 = \frac{c}{d}$, where $a, b, c, d \in \mathbb{Z}$ and $b, d \neq 0$
    • Multiplying gives $r_1 \cdot r_2 = \frac{a \cdot c}{b \cdot d}$
    • Since integers are closed under multiplication, $ac \in \mathbb{Z}$ and $bd \in \mathbb{Z}$, with $bd \neq 0$
    • Therefore, $\frac{ac}{bd}$ fits the definition of a rational number

• Problem 32: Closure Property of Irrational Numbers under Addition

  • Statement: "Any irrational number added to another irrational number will always result in an irrational number."
  • Truth Value: False
  • Explanation: Irrational numbers are not closed under addition.
  • Counterexample:
    • Select irrational numbers $\sqrt{2} \in \mathbb{I}$ and $-\sqrt{2} \in \mathbb{I}$
    • Add them together: $\sqrt{2} + (-\sqrt{2}) = 0$
    • The sum $0$ is a rational number ($0 = \frac{0}{1} \in \mathbb{Q}$), not an irrational number
    • Alternative Counterexample: $(3 + \sqrt{5}) + (2 - \sqrt{5}) = 5$, which is rational