The Main Idea 

• Definition of Square Root:
  • A square root of [latex]a[/latex] is any number [latex]x[/latex] such that [latex]x^2 = a[/latex]
  • For positive [latex]a[/latex], there are always two square roots: positive and negative
• Principal Square Root:
  • Denoted by [latex]\sqrt{a}[/latex]
  • The non-negative square root of a number
  • What calculators provide when computing square roots
• Radical Expression:
  • [latex]\sqrt{a}[/latex] is called a radical expression
  • [latex]\sqrt{}[/latex] symbol is the radical
  • [latex]a[/latex] under the radical is the radicand

The Main Idea

• Simplifying Square Roots:
  • Rewrite as a rational number times the square root of a number with no perfect square factors
• Product Rule for Square Roots:
  • For non-negative real numbers [latex]a[/latex] and [latex]b[/latex]: [latex]\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}[/latex]
• Perfect Squares:
  • Integers that are squares of other integers (e.g., 16, 25, 36)
• Simplification Process:
  • Factor the radicand into perfect square and non-perfect square parts
  • Apply the product rule
  • Simplify
• Variables in Square Roots:
  • [latex]\sqrt{x^2} = |x|[/latex]
  • Even powers can be simplified, odd powers leave one factor under the radical

The Main Idea

• Quotient Rule for Square Roots:
  • For non-negative real numbers [latex]a[/latex] and [latex]b[/latex], where [latex]b \neq 0[/latex]: [latex]\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}[/latex]
• Application of the Quotient Rule:
  • Separate the square root of a fraction into the quotient of two square roots
  • Simplify numerator and denominator separately
• Simplification Process:
  • Apply the quotient rule
  • Simplify each radical expression
  • Combine results

The Main Idea

• Adding and Subtracting Square Roots:
  • Only possible with identical radicands
  • Combine coefficients of like terms
• Simplifying Radical Expressions:
  • Simplify each term before combining
  • Factor out perfect squares from radicands
• Rationalizing Denominators:
  • Remove radicals from the denominator
  • Use the identity property of multiplication
• Methods for Rationalizing:
  • Single term denominator: Multiply by [latex]\sqrt{c}[/latex] for [latex]b\sqrt{c}[/latex]
  • Complex denominator: Use the conjugate for [latex]a + b\sqrt{c}[/latex]
• Conjugates:
  • For [latex]a + \sqrt{b}[/latex], the conjugate is [latex]a - \sqrt{b}[/latex]
  • Product of a term and its conjugate: [latex](a + \sqrt{b})(a - \sqrt{b}) = a^2 - b[/latex]

The Main Idea

• nth Roots:
  • [latex]\sqrt[n]{a}[/latex] is a number that, when raised to the nth power, equals [latex]a[/latex]
  • [latex]n[/latex] is called the index of the radical
• Principal nth Root:
  • Denoted as [latex]\sqrt[n]{a}[/latex]
  • Has the same sign as [latex]a[/latex]
  • When raised to the nth power, equals [latex]a[/latex]
• Rational Exponents:
  • Alternative notation for radical expressions
  • General form: [latex]a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}[/latex]
• Properties of Rational Exponents:
  • Follow the same rules as integer exponents
  • Can simplify expressions by combining like bases
• Conversion between Radicals and Rational Exponents:
  • [latex]\sqrt[n]{x} = x^{\frac{1}{n}}[/latex]
  • [latex]\sqrt[n]{x^m} = x^{\frac{m}{n}}[/latex]