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Surds – Expanding and Rationalising
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Expanding brackets with surds
Expanding single bracket with integer and surd
\[ \sqrt{3}(4 + \sqrt{3}) \]
Expand a bracket where a surd multiplies an integer plus a surd.
Expanding single bracket – surd by two surds
\[ \sqrt{2}(\sqrt{3} + \sqrt{5}) \]
Expand a bracket where a surd multiplies the sum of two surds.
Expanding single bracket with simplification
\[ \sqrt{2}(3 + \sqrt{8}) \]
Expand a bracket where the result requires simplification.
Expanding single bracket with coefficient
\[ 2\sqrt{3}(5 – 3\sqrt{3}) \]
Expand a bracket where a multiple of a surd multiplies a binomial.
Expanding double brackets – same surd
\[ (2 + \sqrt{3})(4 + \sqrt{3}) \]
Expand two brackets where both contain the same surd.
Expanding double brackets – different surds
\[ (1 + \sqrt{2})(3 + \sqrt{5}) \]
Expand two brackets containing different surds.
Expanding double brackets with simplification
\[ (1 + \sqrt{3})(2 + \sqrt{12}) \]
Expand brackets where the result requires simplification.
Squaring a bracket with a surd – addition
\[ (3 + \sqrt{2})^2 \]
Square an expression of the form (a + √b)².
Squaring a bracket with a surd – subtraction
\[ (5 – \sqrt{3})^2 \]
Square an expression of the form (a – √b)².
Difference of two squares with integer and surd
\[ (4 + \sqrt{3})(4 – \sqrt{3}) \]
Expand brackets that form a difference of two squares pattern.
Difference of two squares with two surds
\[ (\sqrt{5} + \sqrt{2})(\sqrt{5} – \sqrt{2}) \]
Expand brackets with two surds that form a difference of two squares.
Rationalising single surd denominators
Rationalising a unit fraction
\[ \frac{1}{\sqrt{2}} \]
Rewrite a fraction with a single surd denominator so the denominator is rational.
Rationalising with integer numerator
\[ \frac{6}{\sqrt{3}} \]
Rationalise when there is an integer in the numerator.
Rationalising with surd numerator
\[ \frac{\sqrt{3}}{\sqrt{2}} \]
Rationalise when there is a surd in the numerator.
Rationalising denominator a√b
\[ \frac{4}{3\sqrt{2}} \]
Rationalise when the denominator is a multiple of a surd.
Rationalising with simplification required
\[ \frac{2}{\sqrt{8}} \]
Rationalise where the surd needs simplifying as part of the process.
Rationalising with surd expression numerator
\[ \frac{3\sqrt{5}}{2\sqrt{3}} \]
Rationalise when both numerator and denominator contain surd expressions.
Rationalising two-term denominators
Identifying the conjugate
\[ 3 + \sqrt{2} \rightarrow \, ? \]
Identify the conjugate of a surd expression.
Rationalising (a + √b) denominator
\[ \frac{1}{3 + \sqrt{2}} \]
Rationalise by multiplying by the conjugate when denominator is integer plus surd.
Rationalising (a + √b) with integer numerator
\[ \frac{5}{2 + \sqrt{3}} \]
Rationalise with an integer numerator and conjugate method.
Rationalising (a – √b) denominator
\[ \frac{6}{5 – \sqrt{7}} \]
Rationalise when the denominator has a minus sign before the surd.
Rationalising (√a + √b) denominator
\[ \frac{1}{\sqrt{5} + \sqrt{2}} \]
Rationalise when the denominator contains two different surds.
Rationalising with surd numerator and two-term denominator
\[ \frac{\sqrt{3}}{1 + \sqrt{2}} \]
Rationalise when there is a surd in the numerator.
Rationalising with binomial numerator
\[ \frac{3 + \sqrt{5}}{2 + \sqrt{5}} \]
Rationalise when both numerator and denominator are binomials with surds.
Complex rationalising with simplification
\[ \frac{2 + \sqrt{8}}{3 + \sqrt{2}} \]
Rationalise where surds need simplifying during the process.
Special cases
Writing in the form √a
\[ 3\sqrt{5} = \sqrt{?} \]
Rewrite a multiple of a surd as a single surd.
Writing in the form a + b√c
\[ \frac{8 + \sqrt{12}}{2} \]
Simplify a fraction to express in standard surd form.
Surds in equations
\[ x^2 = 18 \]
Solve simple equations giving answers as simplified surds.
Comparing surds
\[ 3\sqrt{2} \text{ vs } 2\sqrt{5} \]
Compare the size of two surd expressions without a calculator.
Showing a result is rational
\[ (2 + \sqrt{3})(2 – \sqrt{3}) \]
Demonstrate that a surd expression simplifies to a rational number.
Finding unknown values from surd expressions
\[ (5 + \sqrt{3})^2 = a + b\sqrt{3} \]
Expand a surd expression and identify integer and surd coefficients.
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