Surds, exact calculations and rationalising

(a) $4\sqrt{3}$

1. Use the largest square factor: $48 = 16 \times 3$, so $\sqrt{48} = \sqrt{16}\sqrt{3} = 4\sqrt{3}$.

• B1 The correct answer, $4\sqrt{3}$.

(b) $7$

1. $$9 - 3\sqrt{2} + 3\sqrt{2} - 2 = 7$$

• M1 Four terms with at least three correct, or using the difference of two squares $9 - 2$.
• A1 The correct answer, $7$.

(c) $2\sqrt{3}$

1. Multiply top and bottom by $\sqrt{3}$.
   $$\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$$

• M1 Multiplying top and bottom by $\sqrt{3}$.
• A1 The correct answer, $2\sqrt{3}$.

(a) $5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$

1. $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ and $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$.
2. $5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$.

• M1 Simplifying one surd correctly: $5\sqrt{3}$ or $2\sqrt{3}$.
• A1 Both simplified and added to $7\sqrt{3}$.

(b) $12$

1. $\sqrt{18 \times 8} = \sqrt{144} = 12$.

• B1 The correct answer, $12$.

(a) $9 + 4\sqrt{5}$

1. $$(2 + \sqrt{5})(2 + \sqrt{5}) = 4 + 2\sqrt{5} + 2\sqrt{5} + 5 = 9 + 4\sqrt{5}$$

• M1 Expanding to four terms with at least three correct.
• A1 $9 + 4\sqrt{5}$.

(b) $3 + \sqrt{5}$

1. Multiply top and bottom by $3 + \sqrt{5}$.
   $$\frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{4(3 + \sqrt{5})}{9 - 5}$$
2. $$= \frac{4(3 + \sqrt{5})}{4} = 3 + \sqrt{5}$$

• M1 Multiplying top and bottom by $3 + \sqrt{5}$.
• M1 Simplifying the denominator to 4.
• A1 The correct answer, $3 + \sqrt{5}$.

(a) $\sqrt{11} + 2$

1. $$\sqrt{11}+2$$
2. $$7(\sqrt{11}+2)/(11-4)$$
3. Multiply numerator and denominator by the conjugate $\sqrt{11} + 2$.
4. The denominator becomes ($\sqrt{11}$)$^{2}$ $-$ $2^{2}$ = 11 $-$ 4 = 7.
5. Cancel the factor of 7 to obtain $\sqrt{11} + 2$.

• P1 Establishing $\sqrt{11}+2$ or an equivalent valid method.
• P1 Establishing $7(\sqrt{11}+2)/(11-4)$ or an equivalent valid method.
• A1 Correct answer: $\sqrt{11} + 2$

(a) $\sqrt{7}/2$

1. $$\frac{\sqrt{7}-\sqrt{3}}{4}$$
2. $$\frac{\sqrt{7}+\sqrt{3}}{4}$$
3. $$(\sqrt{7}-\sqrt{3})/4+(\sqrt{7}+\sqrt{3})/4$$
4. Rationalise each fraction using its conjugate. Their common denominator is 7 $-$ 3 = 4.
5. The sum is ($\sqrt{7}$ $-$ $\sqrt{3}$)/4 + ($\sqrt{7}$ + $\sqrt{3}$)/4 = $2\sqrt{7}$/4.
6. Simplify to $\sqrt{7}$/2.

• M1 Establishing $\frac{\sqrt{7}-\sqrt{3}}{4}$ or an equivalent valid method.
• M1 Establishing $\frac{\sqrt{7}+\sqrt{3}}{4}$ or an equivalent valid method.
• M1 Establishing $(\sqrt{7}-\sqrt{3})/4+(\sqrt{7}+\sqrt{3})/4$ or an equivalent valid method.
• A1 Correct answer: $\sqrt{7}/2$

(a) $\frac{27 + 6\sqrt{6}}{19}$

1. $$3\sqrt{3}/(3\sqrt{3}-2\sqrt{2})$$
2. $$\frac{27+6\sqrt{6}}{19}$$
3. Simplify the surds: $\sqrt{75}$ $-$ $\sqrt{12}$ = $5\sqrt{3}$ $-$ $2\sqrt{3}$ = $3\sqrt{3}$, and $\sqrt{27}$ $-$ $\sqrt{8}$ = $3\sqrt{3}$ $-$ $2\sqrt{2}.$
4. Multiply numerator and denominator by the conjugate $3\sqrt{3}$ + $2\sqrt{2}.$
5. The numerator becomes 27 + $6\sqrt{6}$ and the denominator is 27 $-$ 8 = 19.
6. The simplified exact answer is (27 + $6\sqrt{6}$)/19.

• M1 Establishing $3\sqrt{3}/(3\sqrt{3}-2\sqrt{2})$ or an equivalent valid method.
• M1 Establishing $\frac{27+6\sqrt{6}}{19}$ or an equivalent valid method.
• A1 Correct answer: $\frac{27 + 6\sqrt{6}}{19}$

(a) $22$ cm²

1. Use the difference of squares.
   $$(5+\sqrt{3})(5-\sqrt{3})$$
2. Therefore $22$ cm².

• M1 Use the difference of squares.
• A1 Correct answer: $22$ cm²

(b) $20$ cm

1. Add both lengths before doubling.
   $$2(5+\sqrt{3}+5-\sqrt{3})$$
2. Therefore $20$ cm.

• M1 Add both lengths before doubling.
• A1 Correct answer: $20$ cm

(a) $(4 - \sqrt{3})$ cm

1. Width = area ÷ length.
   $$\frac{17 - \sqrt{3}}{5 + \sqrt{3}} \times \frac{5 - \sqrt{3}}{5 - \sqrt{3}}$$
2. Numerator and denominator:
   $$\frac{85 - 17\sqrt{3} - 5\sqrt{3} + 3}{25 - 3} = \frac{88 - 22\sqrt{3}}{22}$$
3. $$= 4 - \sqrt{3}$$

• P1 Writing the width as $\frac{17 - \sqrt{3}}{5 + \sqrt{3}}$.
• P1 Multiplying top and bottom by $5 - \sqrt{3}$.
• P1 Expanding the numerator to $88 - 22\sqrt{3}$.
• A1 The correct answer, $4 - \sqrt{3}$.

(b) $2(5 + \sqrt{3}) + 2(4 - \sqrt{3}) = 18$

1. $$2(5 + \sqrt{3}) + 2(4 - \sqrt{3}) = 10 + 2\sqrt{3} + 8 - 2\sqrt{3} = 18$$

• M1 Adding two lengths and two widths.
• A1 Showing the surds cancel to give 18.