Multiplying and expanding surds questions with answers

Questions

1. Question 1Multiply two surds

   Simplify $2\sqrt{7} \times 3\sqrt{14}$.

   Show the worked solution

   1. Step 1: Multiply the numbers in front, and the numbers under the roots$2\sqrt{7} \times 3\sqrt{14} = 6\sqrt{98}$
   2. Step 2: Simplify the surd: take out the biggest square factor, keeping the number in front$6\sqrt{98} = 6 \times 7\sqrt{2}$$= 42\sqrt{2}$
2. Question 2Expand one bracket

   Expand and simplify $\sqrt{3}(-4 + 2\sqrt{3})$.

   Show the worked solution

   1. Step 1: Multiply each term in the bracket by the surd outside$\sqrt{3} \times (-4) = -4\sqrt{3}$$\sqrt{3} \times 2\sqrt{3} = 6$
   2. Step 2: Collect the terms$\sqrt{3}(-4 + 2\sqrt{3}) = 6 - 4\sqrt{3}$
3. Question 3Expand two brackets

   Expand and simplify $(1 - \sqrt{7})(4 + 2\sqrt{7})$.

   Show the worked solution

   1. Step 1: Multiply every term in the first bracket by every term in the secondFirst: $1 \times 4 = 4$Outer: $1 \times 2\sqrt{7} = 2\sqrt{7}$Inner: $(-\sqrt{7}) \times 4 = -4\sqrt{7}$Last: $(-\sqrt{7}) \times 2\sqrt{7} = -14$
   2. Step 2: Collect the whole numbers and the surds$(1 - \sqrt{7})(4 + 2\sqrt{7}) = -10 - 2\sqrt{7}$
4. Question 4Squares and conjugates

   Expand and simplify $(3 - \sqrt{5})^2$.

   Show the worked solution

   1. Step 1: (a + b)² = a² + 2ab + b²: or multiply every term by every termFirst: $3 \times 3 = 9$Outer: $3 \times (-\sqrt{5}) = -3\sqrt{5}$Inner: $(-\sqrt{5}) \times 3 = -3\sqrt{5}$Last: $(-\sqrt{5}) \times (-\sqrt{5}) = 5$
   2. Step 2: Collect the whole numbers and the surds$(3 - \sqrt{5})^2 = 14 - 6\sqrt{5}$