Simplifying and operating with surds
This optional practice goes beyond the core real-number investigation. Use exact forms and simplify the roots before combining terms
- Extract square factors Find the largest square factor in the radicand, take its square root outside, then multiply by the original coefficient \(\displaystyle \sqrt{48}=4\sqrt3\)
- Combine only matching roots After simplification, add or subtract coefficients of the same square root. Keep unlike roots and rational terms separate \(\displaystyle 5\sqrt7-2\sqrt7=3\sqrt7\)
- Subtract every term in brackets A minus sign before brackets changes every sign inside. A negative coefficient is allowed \(\displaystyle a-(b-c)=a-b+c\)
- Simplify products again Multiply coefficients and radicands separately, then extract any remaining square factors \(\displaystyle \sqrt6\times\sqrt{15}=3\sqrt{10}\)
Worked example
Simplify the expression, including its rational and unlike-root terms
- \(\displaystyle 3+2\sqrt2-\sqrt5-1+4\sqrt2\)Subtract both terms in the bracket
- \(\displaystyle (3-1)+(2+4)\sqrt2-\sqrt5\)Group rational terms and matching roots; keep √5 separate
- \(\displaystyle 2+6\sqrt2-\sqrt5\)The rational term is 2; the coefficients of √2 and √5 are 6 and −1.
Common mistakes
Difficulty levels
Questions about this topic
Are surd operations core Year 9 content?+
This topic is optional practice. VCAA Level 10A explicitly includes surd operations; the core Year 9 topic here focuses on real numbers and exact root positions
Why keep unlike roots separate?+
Only matching square-root parts act as like terms. Simplify first to find whether roots match
Can the coefficient be negative or fractional?+
Yes. Keep that rational coefficient exact, including its sign
Must every correct numerical value be the requested form?+
No. A question asking for the fully simplified form requires square-free radicands. An equivalent form with a square factor still inside the root is unfinished
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 6: Indices and surds