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Surds – simplifying and operations
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1. Foundational skills
Recognising square numbers
\[ \text{Example: } 16, 25 \]
Identify which numbers are perfect squares from a list.
Finding the largest square factor
\[ \text{Example: } 72 \rightarrow 36 \]
Find the largest perfect square that divides exactly into a given number.
2. Simplifying single surds
Simplifying surds with factor of 4
\[ \sqrt{8} \rightarrow 2\sqrt{2} \]
Write a surd in its simplest form when the number has 4 as a factor.
Simplifying surds with factor of 9
\[ \sqrt{45} \rightarrow 3\sqrt{5} \]
Write a surd in its simplest form when the number has 9 as a factor.
Simplifying surds with factor of 16
\[ \sqrt{48} \rightarrow 4\sqrt{3} \]
Write a surd in its simplest form when the number has 16 as a factor.
Simplifying surds with factor of 25
\[ \sqrt{75} \rightarrow 5\sqrt{3} \]
Write a surd in its simplest form when the number has 25 as a factor.
Simplifying surds with factor of 36
\[ \sqrt{72} \rightarrow 6\sqrt{2} \]
Write a surd in its simplest form when the number has 36 as a factor.
Simplifying surds with larger square factors
\[ \sqrt{98} \rightarrow 7\sqrt{2} \]
Write a surd in its simplest form when the number has 49 or larger square factors.
Recognising surds in simplest form
\[ \text{Example: } \sqrt{15} \]
Decide whether a surd can be simplified further.
Simplifying multiples of surds
\[ 3\sqrt{12} \rightarrow 6\sqrt{3} \]
Write a multiple of a surd in its simplest form.
3. Multiplying and dividing surds
Multiplying two simple surds
\[ \sqrt{2} \times \sqrt{3} \rightarrow \sqrt{6} \]
Multiply two surds together using the multiplication rule.
Multiplying surds that require simplifying
\[ \sqrt{6} \times \sqrt{3} \rightarrow 3\sqrt{2} \]
Multiply two surds where the result needs to be simplified.
Multiplying a surd by an integer
\[ 5 \times \sqrt{3} \rightarrow 5\sqrt{3} \]
Multiply an integer by a surd.
Multiplying surd expressions
\[ 2\sqrt{3} \times 4\sqrt{5} \rightarrow 8\sqrt{15} \]
Multiply two expressions of the form $a\sqrt{b} \times c\sqrt{d}$.
Multiplying surd expressions with simplification
\[ 2\sqrt{6} \times 3\sqrt{2} \rightarrow 12\sqrt{3} \]
Multiply two surd expressions where the result needs to be simplified.
Squaring a surd
\[ (\sqrt{7})^2 \rightarrow 7 \]
Square a surd to get an integer.
Squaring a multiple of a surd
\[ (3\sqrt{2})^2 \rightarrow 18 \]
Square an expression of the form $a\sqrt{b}$.
Dividing two simple surds
\[ \sqrt{15} \div \sqrt{3} \rightarrow \sqrt{5} \]
Divide one surd by another using the division rule.
Dividing surds with simplification
\[ \sqrt{72} \div \sqrt{2} \rightarrow 6 \]
Divide surds where the result needs simplifying.
Dividing surd expressions
\[ 12\sqrt{10} \div 4\sqrt{2} \rightarrow 3\sqrt{5} \]
Divide expressions of the form $a\sqrt{b} \div c\sqrt{d}$.
4. Adding and subtracting surds
Adding like surds
\[ 3\sqrt{5} + 7\sqrt{5} \rightarrow 10\sqrt{5} \]
Add two surds with the same number under the root sign.
Subtracting like surds
\[ 9\sqrt{3} – 4\sqrt{3} \rightarrow 5\sqrt{3} \]
Subtract surds with the same number under the root sign.
Adding surds after simplification
\[ \sqrt{12} + \sqrt{27} \rightarrow 5\sqrt{3} \]
Add two surds that need to be simplified first to reveal like terms.
Subtracting surds after simplification
\[ \sqrt{75} – \sqrt{12} \rightarrow 3\sqrt{3} \]
Subtract two surds that need to be simplified first.
Mixed addition and subtraction of surds
\[ \sqrt{48} + \sqrt{12} – \sqrt{27} \rightarrow 3\sqrt{3} \]
Add and subtract multiple surds that simplify to like terms.
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