Surds

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Notes

What is a Surd?

A **surd** is the square root of a non-square integer, e.g. $\sqrt{5}$ .
Surds allow exact answers (e.g. $5\sqrt{2}$ instead of 7.07...).

Multiplying and Dividing Surds

Multiplying: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}, e.g$ . $\sqrt{3} \times \sqrt{5} = \sqrt{15}$ .
Dividing: $\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}, e.g$ . $\sqrt{21} \div \sqrt{7} = \sqrt{3}$ .
Factorising: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}, e.g$ . $\sqrt{35} = \sqrt{5} \times \sqrt{7}$ .

Adding and Subtracting Surds

Only add/subtract **like surds** (same number under root), e.g. $3\sqrt{5} + 8\sqrt{5} = 11\sqrt{5}$ .
Unlike surds cannot be combined, e.g. $2\sqrt{3} + 4\sqrt{6}$ stays as is.
Do not add numbers under square roots: $\sqrt{9} + \sqrt{4} = 3+2=5$ , $not \sqrt{13}$ .

Simplifying Surds

Factorise the number using the largest square factor, e.g. $\sqrt{48} = \sqrt{16\times 3} = 4\sqrt{3}$ .
Simplify multiple surds separately then collect like terms, e.g. $\sqrt{32} + \sqrt{8} = 4\sqrt{2} + 2\sqrt{2} = 6\sqrt{2}$ .
Expand double brackets like algebra, then simplify using $(\sqrt{a})^{2} = a$ .

Rationalising Simple Denominators

If denominator is a surd, multiply numerator and denominator by that surd.
Example: $a/\sqrt{b} = (a/\sqrt{b})\times (\sqrt{b}/\sqrt{b}) = a\sqrt{b}/b$ .
This removes the surd from the denominator because $\sqrt{b} \times \sqrt{b} = b$ .

Rationalising Harder Denominators

If denominator is $a + \sqrt{b}$ , multiply by its conjugate a – $\sqrt{b}$ .
Use difference of squares: (a+√b)(a–√b) $= a^{2}$ – b.
Example: $2/(3+\sqrt{5}) =$ 2(3–√5)/(9–5) = (6–2√5)/4 = (3–√5)/2.

Key Exam Tips

If calculator gives a surd, keep it in surd form throughout working.
After rationalising, check denominator has no surd left.
Always simplify surds fully before adding/subtracting.

Multiplying Surds

Rationalising Denominator (Simple)

Rationalising Denominator (Harder)

Simplifying Surds

Practice questions

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1.Simplify $\sqrt{32} + \sqrt{98}$ .
Easy
- A $\sqrt{130}$
- B $11\sqrt{2}$
- C $9\sqrt{2}$
2.Rationalise the denominator of $12/\sqrt{3}$ .
Easy
- A $4\sqrt{3}$
- B $12\sqrt{3}/3$
- C $4\sqrt{3}/3$
- D $\frac{12}{3}$
3.Simplify $\sqrt{20} + \sqrt{80}$ , giving your answer in the form $a\sqrt{5}$ .
Medium
- A $10\sqrt{5}$
- B $6\sqrt{5}$
- C $2\sqrt{5}$
- D $8\sqrt{5}$
4.Show that $\sqrt{45} + \sqrt{20} = 5\sqrt{5}$ . Which step is correct?
Medium
- A $\sqrt{45} = 3\sqrt{5}, \sqrt{20} = 2\sqrt{5}$ , $sum = 5\sqrt{5}$
- B $\sqrt{45} = 9\sqrt{5}, \sqrt{20} = 4\sqrt{5}$ , $sum = 13\sqrt{5}$
- C $\sqrt{45} = 5\sqrt{5}, \sqrt{20} = 2\sqrt{5}$ , $sum = 7\sqrt{5}$
- D $\sqrt{45} = 3\sqrt{5}, \sqrt{20} = 4\sqrt{5}$ , $sum = 7\sqrt{5}$
5.Rationalise the denominator of $10/\sqrt{6}$ .
Medium
- A $(10\sqrt{6})/6$
- B $(5\sqrt{6})/3$
- C $(10\sqrt{6})/3$
- D $5\sqrt{6}$
6.Simplify $\sqrt{6} \times \sqrt{3}$ .
Medium
- A $3\sqrt{2}$
- B $\sqrt{18}$
- C $3\sqrt{3}$
- D $\sqrt{9}$
7.Simplify fully by rationalising the denominator: $20/\sqrt{5}$ .
Medium
- A $4\sqrt{5}$
- B $20\sqrt{5}/5$
- C $4\sqrt{5}/5$
- D $5\sqrt{5}$
8.Simplify fully: $\sqrt{200}$ .
Medium
- A $10\sqrt{2}$
- B $20\sqrt{10}$
- C $100\sqrt{2}$
- D $10\sqrt{10}$

Summary

## Surds - Night Before Exam Summary ### What is a Surd? - A surd is the square root of a non-square integer (e.g., √2, √5). - Using surds gives exact answers. ### Simplifying Surds - **Multiply:** √a × √b = √(ab) - **Divide:** √a ÷ √b = √(a/b) - **Factorise:** √(ab) = √a × √b - **Add/Subtract:** Only combine like surds (e.g., 3√5 + 8√5 = 11√5). - **Simplify:** Factor out the largest square number. E.g., √48 = √(16×3) = 4√3. - **Double brackets:** Expand like algebra, then simplify using (√a)² = a. ### Rationalising Denominators - **Simple denominator (single surd):** Multiply numerator and denominator by that surd. - E.g., a/√b = (a√b)/b - **Harder denominator (a + √b):** Multiply by the conjugate (a - √b). - Use difference of squares: (a+√b)(a-√b) = a² - b. - Simplify the result. ### Key Tips - Keep answers in surd form throughout working to avoid rounding errors. - If denominator still has a surd after rationalising, check your work. - Common square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100. ### Example Simplify √54 - √24: √54 = √(9×6) = 3√6, √24 = √(4×6) = 2√6, so 3√6 - 2√6 = √6.

Key terms

Surd: The square root of a non-square integer, used to express exact values.
Rationalising the denominator: Rewriting a fraction so that the denominator no longer contains a surd, by multiplying numerator and denominator by an appropriate expression.
Conjugate: For an expression a + √b, the conjugate is a - √b; used to rationalise denominators with two terms.
Like surds: Surds with the same number under the square root, e.g., 3√5 and 8√5 can be added/subtracted.
Difference of two squares: The identity (a+b)(a-b) = a² - b², used to rationalise denominators of the form a + √b.

Slides

What is a Surd?

A surd is the square root of a non-square integer, e.g. $\sqrt{2}, \sqrt{5}$ .
Surds allow you to leave answers in exact form (e.g. $5\sqrt{2}$ instead of 7.07...).
You can multiply and divide surds: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}, \sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}$ .

Which of the following is a surd?

$\sqrt{25}$
$\sqrt{7}$
$\sqrt{16}$
$\sqrt{9}$

Simplifying Surds

To simplify a surd, factorise the number using a square number factor.
Use the largest square factor: e.g. $\sqrt{48} = \sqrt{16\times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$ .
Simplify multiple surds separately then collect like terms.

Simplify $\sqrt{50} = ____√____$

Adding and Subtracting Surds

You can only add or subtract multiples of the same surd (like terms).
Example: $3\sqrt{5} + 8\sqrt{5} = 11\sqrt{5}$ .
You cannot add or subtract numbers under the square root: $\sqrt{9} + \sqrt{4} \ne \sqrt{13}$ .

Which of these are true?

$\sqrt{3} + \sqrt{3} = \sqrt{6}$
$2\sqrt{7} + 5\sqrt{7} = 7\sqrt{7}$
$\sqrt{12} + \sqrt{3} = 3\sqrt{3}$
$\sqrt{2} + \sqrt{3} = \sqrt{5}$

Rationalising Denominators (Simple)

If the denominator is a single surd, multiply numerator and denominator by that surd.
Example: $4/\sqrt{5} = (4/\sqrt{5})\times (\sqrt{5}/\sqrt{5}) = 4\sqrt{5}/5$ .
This removes the surd from the denominator.

Rationalise $10/\sqrt{5}$ . What is the simplified form?

$10\sqrt{5}/5$
$2\sqrt{5}$
$\sqrt{5}/2$
$10\sqrt{5}$

Rationalising Denominators (Harder)

If denominator is $a + \sqrt{b}$ , multiply numerator and denominator by $a - \sqrt{b}$ .
Use the difference of two squares: $(a+\sqrt{b})(a-\sqrt{b}) = a^{2} - b$ .
Example: $4/(6-\sqrt{2}) = 4(6+\sqrt{2})/(36-2) = (24+4\sqrt{2})/34 = (12+2\sqrt{2})/17$ .

Rationalise $2/(3+\sqrt{5}) = (____ - \sqrt{5})/____$

Expanding Brackets with Surds

Expand double brackets just like algebra, then simplify using $\sqrt{a} \times \sqrt{a} = a$ .
Example: $(6-\sqrt{2})(6+4\sqrt{2}) = 36 + 24\sqrt{2} - 6\sqrt{2} - 8 = 28 + 18\sqrt{2}$ .
Collect like terms and simplify surds where possible.

Expand $(1+\sqrt{3})(2-\sqrt{3})$ . What is the result?

$2+\sqrt{3}-3$
$-1+\sqrt{3}$
$2-\sqrt{3}+2\sqrt{3}-3$
$-1+\sqrt{3}$

Worked Example: Simplifying

Simplify $\sqrt{54} - \sqrt{24}$ .
$\sqrt{54} = \sqrt{9\times 6} = 3\sqrt{6}, \sqrt{24} = \sqrt{4\times 6} = 2\sqrt{6}$ .
$So \sqrt{54} - \sqrt{24} = 3\sqrt{6} - 2\sqrt{6} = \sqrt{6}$ .

Simplify $\sqrt{45} + \sqrt{20} = ____\sqrt{5}$

Worked Example: Rationalising

Write $4/(6-\sqrt{2})$ in the form $p + q\sqrt{r}$ .
Multiply by $(6+\sqrt{2})/(6+\sqrt{2}): = 4(6+\sqrt{2})/(36-2) = (24+4\sqrt{2})/34 = (12+2\sqrt{2})/17$ .
So $p = \frac{12}{17}, q = \frac{2}{17}, r = 2$ .

Rationalise $1/(\sqrt{2}+1)$ . What is the simplified form?

$\sqrt{2}-1$
$\sqrt{2}+1$
$1-\sqrt{2}$
$2-\sqrt{2}$

Key Exam Tip

If your calculator gives a surd, keep it as a surd throughout your working.
This avoids losing accuracy.
If your final answer still has a surd in the denominator, check your working!

Which of these are true?

$\sqrt{12} = 2\sqrt{3}$
$\sqrt{8} + \sqrt{2} = \sqrt{10}$
$5/\sqrt{5} = \sqrt{5}$
$1/(\sqrt{3}-1) = (\sqrt{3}+1)/2$

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