Study guides › 0580 › Number Maths4U Original · Learn, check, explain
Simplifying and combining surds — 0580
Extract square factors and combine only like surds.
Extended · Syllabus E1.18
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods For positive values, \(\displaystyle \sqrt{a^{2}d}=a\sqrt{d}\) when a>0. Extract the largest square factor so the remaining integer under the root has no square factor. Like surds combine by adding their coefficients.
Multiplication can remove a root: \(\displaystyle \sqrt{d}\times \sqrt{d}=d\). Addition does not distribute over square roots; \(\displaystyle \sqrt{u+v}\) is generally different from \(\displaystyle \sqrt{u}+\sqrt{v}\). Keep an exact surd when requested rather than replacing it by a decimal.
Check the conditions Use the stated units and precision. Show the method as well as the final answer. Three worked examples Example 1 · Simplifying and combining surds Keep all values exact. Each square root denotes its nonnegative real value.
Write \(\displaystyle \sqrt{80}\) as \(\displaystyle k\sqrt{5}\). Find k. Find the coefficient of \(\displaystyle \sqrt{5}\) in \(\displaystyle 4\sqrt{5}+3\sqrt{5}\). Find \(\displaystyle \sqrt{80} \times \sqrt{45}\). Find the coefficient of \(\displaystyle \sqrt{5}\) in \(\displaystyle 4\sqrt{5}-3\sqrt{5}\). Show the reasoning (a) Extract square factors first, then apply the stated sum, difference or product rule.
\(4\)
Answer: \(4\).
(b) Extract square factors first, then apply the stated sum, difference or product rule.
\(7\)
Answer: \(7\).
(c) Extract square factors first, then apply the stated sum, difference or product rule.
\(60\)
Answer: \(60\).
(d) Extract square factors first, then apply the stated sum, difference or product rule.
\(1\)
Answer: \(1\).
Example 2 · Simplifying and combining surds Keep all values exact. Each square root denotes its nonnegative real value.
Write \(\displaystyle \sqrt{150}\) as \(\displaystyle k\sqrt{6}\). Find k. Find the coefficient of \(\displaystyle \sqrt{6}\) in \(\displaystyle 5\sqrt{6}+4\sqrt{6}\). Find \(\displaystyle \sqrt{150} \times \sqrt{96}\). Find the coefficient of \(\displaystyle \sqrt{6}\) in \(\displaystyle 5\sqrt{6}-4\sqrt{6}\). Show the reasoning (a) Extract square factors first, then apply the stated sum, difference or product rule.
\(5\)
Answer: \(5\).
(b) Extract square factors first, then apply the stated sum, difference or product rule.
\(9\)
Answer: \(9\).
(c) Extract square factors first, then apply the stated sum, difference or product rule.
\(120\)
Answer: \(120\).
(d) Extract square factors first, then apply the stated sum, difference or product rule.
\(1\)
Answer: \(1\).
Example 3 · Simplifying and combining surds Keep all values exact. Each square root denotes its nonnegative real value.
Write \(\displaystyle \sqrt{72}\) as \(\displaystyle k\sqrt{2}\). Find k. Find the coefficient of \(\displaystyle \sqrt{2}\) in \(\displaystyle 6\sqrt{2}+5\sqrt{2}\). Find \(\displaystyle \sqrt{72} \times \sqrt{50}\). Find the coefficient of \(\displaystyle \sqrt{2}\) in \(\displaystyle 6\sqrt{2}-5\sqrt{2}\). Show the reasoning (a) Extract square factors first, then apply the stated sum, difference or product rule.
\(6\)
Answer: \(6\).
(b) Extract square factors first, then apply the stated sum, difference or product rule.
\(11\)
Answer: \(11\).
(c) Extract square factors first, then apply the stated sum, difference or product rule.
\(60\)
Answer: \(60\).
(d) Extract square factors first, then apply the stated sum, difference or product rule.
\(1\)
Answer: \(1\).
Common mistakes Keep intermediate values exact until the final step. Check the answer against the original conditions. Practise with answer checking Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
Next skill: Rationalising a denominator →
About these materials Prepared independently from learning objectives with AI assistance for Maths4U. Mathematical checks accompany the authored models; independent teacher review is pending. Maths4U marking guidance is not an official examination-board mark scheme. Existing questions keep their original sources and results.