Surds — NSW Year 10 Advanced Maths
Exact values like the square root of 2, from simplifying and ordering surds to rationalising denominators. Six subtopics and 120 practice questions, each with a fully worked solution, aligned to NESA outcome MA5-IND-P-02.
Surds in NSW Year 10 Advanced
A surd is a root that cannot be simplified to a rational number, such as \(\sqrt{2}\) or \(\sqrt[3]{16}\). Surds give exact answers where a calculator would only give a rounded decimal. In Year 10 Advanced, students simplify surds, tell rational and irrational numbers apart, and add, subtract, multiply, divide and expand expressions that contain surds.
The six subtopics below begin with simplifying surds and with surds and rational numbers, including recurring decimals as fractions, then move through adding and subtracting, multiplying and dividing, and binomial products, and finish with rationalising the denominator. Surds is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcome MA5‑IND‑P‑02). Exact surd answers appear again in trigonometry, coordinate geometry, quadratic equations and Year 11 Advanced.
Subtopics
Surds: common questions
What is a surd in Year 10 maths?
A surd is an irrational root, such as \(\sqrt{2}\), \(\sqrt{45}\) or \(\sqrt[3]{90}\). Roots of perfect squares or cubes, such as \(\sqrt{121}=11\), are not surds because they are rational.
How do you simplify a surd?
Write the number under the root as a product with the largest possible perfect square, then take its square root out. For example, \(\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\).
Can you add surds like \(\sqrt{2}+\sqrt{3}\)?
No. Only like surds can be added, so \(\sqrt{2}+\sqrt{3}\) stays as it is. Simplify first to check: \(\sqrt{12}+\sqrt{27}=2\sqrt{3}+3\sqrt{3}=5\sqrt{3}\).
How do you rationalise a denominator with two terms?
Multiply the top and bottom by the conjugate, which has the opposite sign in the middle. For example, \(\frac{4}{\sqrt{7}-\sqrt{3}}\times\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}+\sqrt{3}}=\frac{4(\sqrt{7}+\sqrt{3})}{4}=\sqrt{7}+\sqrt{3}\).
Which NESA outcome covers surds?
Surds are Path content in Stage 5 of the NSW Mathematics K–10 Syllabus (2022), under outcome MA5‑IND‑P‑02, which also covers fractional indices. They are expected of students heading for Mathematics Advanced in Years 11 and 12.