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The best approach here is to first rationalise the denominator. We do this by multiplying both top and bottom by an approriate value.

For example to rationalise the denominator for an expression like $\\frac{a}{\\sqrt{b}}$, we multiply numerator and denominator by $\\sqrt{b}$ to get $\\frac{a\\sqrt{b}}{b}$

Similarly to rationalise the denominator for an expression like $\\frac{a+\\sqrt{b}}{c+\\sqrt{d}}$, we multiply numerator and denominator by $c-\\sqrt{d}$ to get $\\frac{(a+\\sqrt{b})(c-\\sqrt{d})}{c^2-d}$

Dont forget that:

$\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}$

$\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}}$

$(a+\\sqrt{b})(a-\\sqrt{b})=a^2-b$

Remember to check that your answer is in its simplest form.

", "name": "Katherine's copy of Surds 3", "statement": "

Express each of the following in the form $\\frac{a+b\\sqrt{c}}{d}$ where a, b, c and d are integers, with a, b, and d having no common factors

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Rationalising Denominator (complex)

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1$\\var{a1}-\\sqrt{\\var{a2}}$ $=$ [[0]]+$\\sqrt{\\var{a2}}$[[1]]

1$\\var{b1}-\\sqrt{\\var{b2}}$ $=$ [[0]]+$\\sqrt{\\var{b2}}$[[1]]

$\\var{c3}$$\\var{c1}-\\sqrt{\\var{c2}}$$=$[[0]]$+$[[1]]$\\sqrt{\\var{c2}}$[[2]]

$\\var{d3}$$\\var{d1}-\\sqrt{\\var{d2}}$ $=$ [[0]]$+$[[1]]$\\sqrt{\\var{d2}}$[[2]]

$\\var{e1}-\\sqrt{\\var{e3}}$$\\var{e2}-\\sqrt{\\var{e3}}$ $=$ [[0]]$+$[[1]]$\\sqrt{\\var{e3}}$[[2]]

$\\var{f1}+\\sqrt{\\var{f3}}$$\\var{f2}-\\sqrt{\\var{f3}}$$ = $ [[0]]$+$[[1]]$\\sqrt{\\var{f3}}$[[2]]

$\\var{g1}+\\sqrt{\\var{g13}}$$\\var{g2}+2\\sqrt{\\var{g3}}$$ =$ [[0]]$+$[[1]]$\\sqrt{\\var{g3}}$[[2]]

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