Surds Flashcards

All 12 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: $\sqrt{40} = 2\sqrt{10}$

What does it mean to rationalise the denominator of a fraction?

Rewrite the fraction so that the denominator is a rational number, with no surd in it.
e.g. $\frac{1}{\sqrt{2}}$ becomes $\frac{\sqrt{2}}{2}$

What are the steps to rationalise a denominator of the form $k/\sqrt{a}$?

e.g. $\frac{6}{\sqrt{3}}$

1. Multiply numerator and denominator by $\sqrt{3}$
2. Denominator becomes rational: $\frac{6\sqrt{3}}{3}$
3. Simplify: $2\sqrt{3}$

What is the conjugate of $a + \sqrt{b}$?

e.g. the conjugate of $3 - \sqrt{5}$

$a - \sqrt{b}$; the conjugate of $3 - \sqrt{5}$ is $3 + \sqrt{5}$.
Their product is rational: $9 - 5 = 4$.

What are the steps to rationalise a denominator of the form $r \pm s\sqrt{a}$?

e.g. $\frac{1+\sqrt{5}}{3-\sqrt{5}}$

1. Multiply numerator and denominator by the conjugate $3+\sqrt{5}$
2. Denominator: $9 - 5 = 4$; numerator: $3 + \sqrt{5} + 3\sqrt{5} + 5 = 8 + 4\sqrt{5}$
3. Simplify fully: $2 + \sqrt{5}$

True or false? Rationalising the denominator changes the value of the fraction.

False. It gives an equal (equivalent) fraction — only the form changes, e.g. $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

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 $9$).