Surds flashcards

Key points

• A surd is a root that is irrational: it cannot be written as a whole number or a fraction.
• $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{5}$ are surds; their decimals go on for ever without repeating.
• Leaving $\sqrt{2}$ as it is keeps the value exact, where any decimal is rounded.

Surd or not?

Key points

• Find the largest square number that divides the number under the root.
• Split the root: $\sqrt{ab}$ = $\sqrt{a}$ × $\sqrt{b}$.
• Take the square root of the square number outside: $\sqrt{25}$ × $\sqrt{3}$ = $5\sqrt{3}$.
• Squares to look for: 4, 9, 16, 25, 36, 49, 64, 81, 100.

Example

Key points

• Multiply under one root: $\sqrt{a}$ × $\sqrt{b}$ = $\sqrt{ab}$.
• Divide under one root: $\sqrt{a}$ ÷ $\sqrt{b}$ = $\sqrt{a÷b}$.
• A root times itself gives the number under it: $\sqrt{a}$ × $\sqrt{a}$ = a.
• Whole numbers multiply with whole numbers, roots with roots: $2\sqrt{3}$ × $5\sqrt{2}$ = $10\sqrt{6}$.

Example

Key points

• Only like surds combine: $3\sqrt{2}$ + $5\sqrt{2}$ = $8\sqrt{2}$, just as 3x + 5x = 8x.
• Unlike surds stay as they are: $\sqrt{2}$ + $\sqrt{3}$ cannot be simplified.
• Simplify each surd first: that often turns unlike surds into like ones.

$\sqrt{50}$ − $\sqrt{8}$

Key points

• Multiply every term in one bracket by every term in the other, as in algebra.
• Collect the whole numbers and the surds separately.
• $(a+\sqrt{b})(a-\sqrt{b})$ = a² − b: the surd terms cancel, leaving a whole number.

Example

Key points

• Rationalising means rewriting the fraction with no surd on the bottom.
• Multiply the top and the bottom by the surd on the bottom.
• $\sqrt{b}$ × $\sqrt{b}$ = b, a whole number; and multiplying by $\frac{\sqrt{b}}{\sqrt{b}}$ is multiplying by 1.
• Simplify at the end if you can.

$\frac{10}{\sqrt{5}}$

Key points

• Multiply the top and the bottom by the conjugate: the bottom with its sign flipped.
• $(b+\sqrt{c})(b-\sqrt{c})$ = b² − c: the middle terms cancel, so no surd is left.
• Expand the top, then simplify by dividing through if every term shares a factor.

$\frac{4}{3+\sqrt{5}}$

Key points

• Exact, or in the form $a\sqrt{b}$, means leave the surd in: a rounded decimal scores nothing.
• Keep surds through the working rather than reaching for the calculator.
• Simplify the surd at the end, so $\sqrt{20}$ becomes $2\sqrt{5}$.
• Pythagoras often gives surd answers.

Diagonal of a 2 cm by 4 cm rectangle

Key points

• Area multiplies the lengths: expand any brackets and simplify.
• Perimeter adds the lengths: collect the whole numbers and the like surds.
• Leave the answer in surd form, with its units.

Rectangle $(3+\sqrt{2})$ cm by $(3-\sqrt{2})$ cm