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NSWMaths AdvancedQuick questions

Year 11: Functions

Quick questions on Surds and index laws for HSC Maths Advanced: the real number system, simplifying surds, surd arithmetic and binomial expansions, rationalising single- and two-term denominators, and the integer index laws

7short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the real number system?
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Mathematics builds its numbers up in layers, each one larger than the last:
What is simplifying a surd?
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A surd is in simplest form when the number under the root sign has no square factor bigger than 11. To simplify, pull out the largest perfect square (4,9,16,25,36,…4, 9, 16, 25, 36, \dots) that divides the number, using the law ab=a b\sqrt{ab} = \sqrt{a}\,\sqrt{b} for non-negative aa and bb:
What is surd arithmetic?
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Once surds are simplified, the ordinary rules of algebra apply, with 2\sqrt{2}, 3\sqrt{3}, and so on treated like the pronumerals xx, yy.
What is rationalising the denominator?
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It is conventional, and almost always required for full marks, to leave no surd in a denominator. Removing it is called rationalising, and there are two cases.
What is rationalise a binomial denominator, stage by stage?
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Stage 1, spot the conjugate. The denominator is 5+1\sqrt{5} + 1. Write down its conjugate, 5−1\sqrt{5} - 1 (same terms, opposite middle sign). Multiplying these two will give (5)2−12(\sqrt{5})^2 - 1^2, a difference of squares with no surd, so the plan is to multiply top and bottom by 5−1\sqrt{5} - 1.
What is the index laws (integer indices)?
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A power ana^n is repeated multiplication, and five laws let you combine powers of the same base. They are the rules you will use to differentiate, to handle exponentials, and to tidy almost any algebraic expression:
What is not taking out the largest square?
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Writing 72=218\sqrt{72} = 2\sqrt{18} is unfinished, because 18=9×218 = 9 \times 2 still has a square factor. Reduce until the number under the root has no square factor bigger than 11.

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