1.1.6 Working with surds and rationalizing denominators

• What are surds?

  Numbers as square roots that can't simplify
• Why are √2 and √5 considered surds?

  They can't be expressed as exact numbers
• Why is √4 not a surd?

  It equals 2, a whole number
• What is the difference between surds and non-surd numbers?

  • Surds: Cannot be simplified to whole numbers or fractions (e.g., √2, √5)
  • Not Surds: Can be simplified to whole numbers or fractions (e.g., √4, √9)
• What is the simplification of $\sqrt{20}$?


  $2\sqrt{5}$
• What is the simplification of $\sqrt{45}$?


  $3\sqrt{5}$
• How does simplifying a surd relate to the concept of whole numbers?

  Simplifying helps express it in whole number terms
• What is the process to simplify a surd?

  Break it down into factors with a perfect square
• What are surds?

  Numbers that can't be made into whole numbers
• What happens when you open a square root box containing a surd?

  You can't make it into a whole number
• What is the process of simplifying a surd?

  • Identify the surd
  • Factor it into simpler components
  • Extract whole numbers from the square root
  • Express the result in simplified form
• How can you simplify $\sqrt{12}$?


  By expressing it as $2\sqrt{3}$
• How does $\sqrt{12}$ simplify to $2\sqrt{3}$?


  It factors to $\sqrt{4 \times 3}$ and simplifies
• Why is $\sqrt{2}$ considered a surd?


  It remains in pieces and isn't whole
• What is the perfect square factor of $\sqrt{12}$?


  $\sqrt{4 \times 3}$
• How is $\sqrt{45}$ simplified?


  It factors to $\sqrt{9 \times 5}$ and simplifies
• What is required to add or subtract surds?

  They need to have the same root
• What should you do if the roots of surds are different?

  Simplify if possible
• What are the simplifications of the surds in the table?

  • $\sqrt{12} \rightarrow 2\sqrt{3}$
  • $\sqrt{45} \rightarrow 3\sqrt{5}$
  • $\sqrt{20} \rightarrow 2\sqrt{5}$
• What is the process to simplify surds?

  • Identify perfect square factors
  • Rewrite the surd using these factors
  • Simplify by taking the square root of the perfect square
• What is the rule for multiplying surds?

  Multiply the numbers inside the square roots
• What is the rule for dividing surds?

  Divide the numbers inside the square roots
• How do you multiply $\sqrt{2} \times \sqrt{3}$?


  $\sqrt{2 \times 3} =$$\sqrt{6}$
• What is the multiplication rule for surds?

  • $\sqrt{a} \times \sqrt{b} =$$\sqrt{a \times b}$
  • Example: $\sqrt{5} \times \sqrt{3} =$$\sqrt{15}$
• What is the simplification of $\sqrt{8} +$$\sqrt{2}$?


  • $\sqrt{8} =$$2\sqrt{2}$
  • Combine: $2\sqrt{2} +$$\sqrt{2} =$$3\sqrt{2}$
• How do you rationalize $\frac{3}{\sqrt{2}}$?


  Multiply by $\sqrt{2}$
• What is the first step in rationalizing a denominator?

  Identify the surd in the denominator
• What does rationalizing the denominator compare to in the study material?

  Getting rid of a monster from a fraction
• What is the first step in simplifying $\sqrt{12}$?


  Cut it into $\sqrt{4 \times 3}$
• What is the result of rationalizing $\frac{3}{\sqrt{2}}$?


  $\frac{3\sqrt{2}}{2}$
• What does it mean to rationalize a denominator?

  To remove the square root from a fraction
• What is the conjugate of a binomial expression?

  Same expression with the sign changed
• What happens to the "monster" after rationalizing the denominator?

  It is eliminated from the bottom
• What is the final result of $\sqrt{8} +$$\sqrt{2}$ after simplification?


  $3\sqrt{2}$
• How do you divide $\frac{\sqrt{8}}{\sqrt{2}}$?


  $\sqrt{\frac{8}{2}} =$$\sqrt{4} =$$2$
• What is the division rule for surds?

  • $\frac{\sqrt{a}}{\sqrt{b}} =$$\sqrt{\frac{a}{b}}$
  • Example: $\frac{\sqrt{12}}{\sqrt{3}} =$$\sqrt{4} =$$2$
• If the denominator is $\sqrt{2}$, what is the magic mirror used?


  $\sqrt{2}$
• If the denominator is $\sqrt{5}$, what should you multiply by to rationalize?


  $\sqrt{5}$
• What is the rationalized form of $\frac{2}{1 + \sqrt{3}}$?


  $- (1 - \sqrt{3})$
• What are the steps to rationalize a denominator?

  1. Identify the surd in the denominator.
  2. Multiply both the numerator and denominator by the conjugate of the denominator.
  3. Simplify both the numerator and denominator.