Surds

• What is a surd?


  A surd is the square root of a non-square integer.
• Why is using surds beneficial in calculations?

  Using surds allows you to leave answers in exact form.
• How would you express the square root of 5 in decimal form?

  Approximately 7.071067811...

  
• How do you multiply surds?

  You can multiply numbers under square roots together.
• What is the result of multiplying $\sqrt{3} \times \sqrt{5}$?


  $\sqrt{15}$
• How do you divide surds?

  You can divide numbers under square roots.
• What is the result of dividing $\sqrt{21} \div \sqrt{7}$?


  $\sqrt{3}$
• How can you factorise surds?

  You can factorise numbers under square roots.
• What is the factorisation of $\sqrt{35}$?


  $\sqrt{5 \times 7}$
• How do you add or subtract surds?

  You can only add or subtract multiples of "like" surds.
• What is the result of $3\sqrt{5} +$$8\sqrt{5}$?


  $11\sqrt{5}$
• What is the result of $7\sqrt{3} - 4\sqrt{3}$?


  $3\sqrt{3}$
• Why can't you add or subtract numbers under square roots directly?


  Because they are not like terms.
• What is the incorrect way to add $9 +$$4$ when considering surds?


  $9 +$$4 =$$13 =$$3.60555...$
• What should you do if your calculator gives you an answer as a surd during an exam?


  Leave the value as a surd throughout the rest of your calculations.
• How do you simplify a surd?

  Separate out a square factor and square root it.

  
• How do you simplify $\sqrt{48}$?


  $\sqrt{16 \times 3} =$$4\sqrt{3}$
• What is the importance of simplifying surds?

  It helps reduce expressions and collect like terms.
• How would you simplify $\sqrt{32} +$$\sqrt{8}$?


  $4\sqrt{2} +$$2\sqrt{2} =$$6\sqrt{2}$
• What is the property used when multiplying double brackets containing surds?

  The property $(a + b)(a - b) =$$a^2 - b^2$ can be used to simplify the expression.
• How do you write $\sqrt{54} - \sqrt{24}$ in the form $p\sqrt{q}$?


  By simplifying both surds separately and collecting like terms.
• What does it mean to rationalise a denominator?

  It means changing a fraction with surds in the denominator into an equivalent fraction with an integer denominator.
• How do you rationalise a denominator that is a surd?

  Multiply the top and bottom by the surd on the denominator.
• What is the first step to rationalise the denominator of $\frac{a}{b}?$ if $b$ is a surd?


  Multiply $\frac{a}{b}$ by $\frac{b}{b}$.
• What happens to the denominator after multiplying by the surd?

  The denominator becomes an integer.
• How do you rationalise a denominator that is an expression containing a surd?

  Multiply the top and bottom by the expression on the denominator, but with the sign changed.

  
• What is the result of rationalising $\frac{2}{1 + \sqrt{3}}$?


  Multiply by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$.
• What is the goal of rationalising the denominator?

  The goal is to remove the surd from the denominator.
• What is the result of expanding the denominator when rationalising?

  It becomes a difference of two squares problem.

  
• How do you simplify the expression after rationalising?

  By cancelling out common factors in the numerator and denominator.
• How would you write $\frac{4}{6 - 2\sqrt{3}}$ in the form $p +$$\frac{q}{r}$?


  By rationalising the denominator and simplifying.
• What should you do if the rationalisation does not remove the surd from the denominator?

  You need to check your working or rethink the expression you are using.