A gardener orders three separate sections of decorative edging with lengths \(\sqrt{45}\), \(\sqrt{20}\) and \(\sqrt5\) metres to surround a newly planted f...
Question 1 Report
A gardener orders three separate sections of decorative edging with lengths \(\sqrt{45}\), \(\sqrt{20}\) and \(\sqrt5\) metres to surround a newly planted flower bed on the allotment. Simplify \(\sqrt{45} - \sqrt{20} + \sqrt5\) fully, giving your answer in the form \(k\sqrt5\). (3)
Answer Details
To add or subtract surds, each one must first be written in terms of the same surd, by taking out the largest perfect square factor from underneath each root.
Simplify each edging length: \(\sqrt{45} = \sqrt{9\times5} = 3\sqrt5\), and \(\sqrt{20} = \sqrt{4\times5} = 2\sqrt5\); \(\sqrt5\) is already in simplest form. [1 mark] Now every term is a multiple of \(\sqrt5\), so they can be combined like ordinary algebraic terms: \(3\sqrt5 - 2\sqrt5 + \sqrt5 = (3-2+1)\sqrt5 = 2\sqrt5\). [2 marks] Comparing with the form \(k\sqrt5\) gives \(k=2\).
Exam tip: surds can only be combined once they share the same number under the root sign - always simplify every surd in an expression before adding or subtracting.
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