Surds Flashcards

All 15 cards in this deck

What are the steps to simplify a single surd?

e.g. simplify $\sqrt{12}$

1. Find the largest square factor: $12 = 4 \times 3$
2. Split the root: $\sqrt{12} = \sqrt{4} \times \sqrt{3}$
3. Root the square factor: $2\sqrt{3}$

What must you do before you can add or subtract two surds?

e.g. $\sqrt{80} - \sqrt{5}$

Simplify each surd so the surd parts are the same, then combine the numbers in front.
$4\sqrt{5} - \sqrt{5} = 3\sqrt{5}$

What is $\sqrt{a} \times \sqrt{b}$ equal to?

e.g. $\sqrt{2} \times \sqrt{8}$

$\sqrt{ab}$
$\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$

True or false? $\sqrt{a} + \sqrt{b} = \sqrt{a+b}$

False. Surds cannot be combined that way; e.g. $\sqrt{9} + \sqrt{16} = 7$ but $\sqrt{25} = 5$.

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

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

1. Write as two brackets and expand all four terms: $4 + 2\sqrt{3} + 2\sqrt{3} + 3$
2. Use $(\sqrt{a})^2 = a$, so $\sqrt{3}\times\sqrt{3}=3$
3. Collect like terms: $7 + 4\sqrt{3}$

What does it mean to give a length or an area in exact form (surd form)?

Leave any square root as a fully simplified surd rather than rounding it to a decimal.
e.g. a square of area $50$ cm$^2$ has side $5\sqrt{2}$ cm.

What does 'rationalise the denominator' mean?

Rewrite the fraction as an equal fraction whose denominator is a rational number (no surd in it).

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

e.g. $\frac{2}{\sqrt{11}}$

1. Multiply numerator and denominator by $\sqrt{11}$
2. Denominator: $\sqrt{11}\times\sqrt{11}=11$
3. Answer: $\frac{2\sqrt{11}}{11}$

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

e.g. $\frac{14}{2+\sqrt{3}}$

1. Multiply numerator and denominator by the conjugate $2-\sqrt{3}$
2. Expand both: $\frac{28-14\sqrt{3}}{4-3}$
3. Simplify: $28-14\sqrt{3}$

What is $(a+\sqrt{b})(a-\sqrt{b})$ equal to?

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

$a^2 - b$, which is rational (the surd terms cancel).
$(2+\sqrt{3})(2-\sqrt{3}) = 4-3 = 1$

True or false? Expanding $(2+\sqrt{3})(2-\sqrt{3})$ gives $4+3=7$

False. $\sqrt{3}\times(-\sqrt{3}) = -3$, not $+3$, so the answer is $4-3=1$.

What should you do first when writing $\frac{8+\sqrt{12}}{5+\sqrt{3}}$ in the form $\frac{a+\sqrt{3}}{b}$?

Simplify $\sqrt{12}$ to $2\sqrt{3}$ so all surds match, then multiply numerator and denominator by the conjugate $5-\sqrt{3}$.

A student simplifies $\sqrt{12}$ as $3\sqrt{2}$. What is the error?

$12 = 4 \times 3$, so $\sqrt{12} = \sqrt{4}\times\sqrt{3} = 2\sqrt{3}$, not $3\sqrt{2}$.

A student rationalises $\frac{14}{2+\sqrt{3}}$ by multiplying the numerator and denominator by $2+\sqrt{3}$. What is the error?

You must multiply by the conjugate $2-\sqrt{3}$; multiplying by $2+\sqrt{3}$ leaves a surd in the denominator.

True or false? To rationalise $\frac{5}{\sqrt{12}}$ you may multiply just the denominator by $\sqrt{12}$.

False. You must multiply the numerator and the denominator by $\sqrt{12}$, or the value of the fraction changes.