Simplifying Surds Flashcards

All 7 cards in this deck

What are the steps to simplify a surd into the form $a\sqrt{b}$?

e.g. simplify $\sqrt{72}$

1. Find the largest square factor: $72 = 36 \times 2$
2. Split the root: $\sqrt{36} \times \sqrt{2}$
3. Root the square factor: $6\sqrt{2}$

What are the steps to multiply two surds and simplify?

e.g. $\sqrt{8} \times \sqrt{6}$

1. Combine under one root: $\sqrt{48}$
2. Take out the largest square factor: $48 = 16 \times 3$
3. Simplify: $4\sqrt{3}$

True or false? $\sqrt{5} \times \sqrt{5} = 5\sqrt{5}$

False. $\sqrt{5} \times \sqrt{5} = 5$ (neither $5\sqrt{5}$ nor $25$).

What are the steps to add or subtract surds?

e.g. $\sqrt{50} + \sqrt{18}$

1. Simplify each surd: $\sqrt{50} = 5\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$
2. Check the surd parts match: both are $\sqrt{2}$
3. Add the coefficients: $8\sqrt{2}$

What are the steps to expand and simplify a squared surd bracket?

e.g. $(2+\sqrt{3})^2$

1. Write as $(2+\sqrt{3})(2+\sqrt{3})$ and expand to four terms: $4 + 2\sqrt{3} + 2\sqrt{3} + \sqrt{3}\times\sqrt{3}$
2. Use $\sqrt{3}\times\sqrt{3} = 3$
3. Collect like terms: $7 + 4\sqrt{3}$

What are the steps to find the exact distance between two points in the form $a\sqrt{b}$?

e.g. $S(5,-4)$ and $T(7,2)$

1. Square the coordinate differences: $(7-5)^2 + (2-(-4))^2 = 2^2 + 6^2$
2. Add them: $4 + 36 = 40$
3. Simplify the root: $\sqrt{40} = 2\sqrt{10}$

True or false? $\sqrt{15}$ is already fully simplified and cannot be written as $a\sqrt{b}$ with $b$ smaller.

True. $15 = 3 \times 5$ has no square factor other than 1, so $\sqrt{15}$ cannot be simplified (it is not $3\sqrt{3}$ or $9$).