Exponents and Surds Worked ExamplesIGCSE Maths 0580

Three different operations on the indices — add, subtract, multiply — and they apply only when the base is the same. The June 2025 examiner report noted that, when multiplying terms like those in (d), dealing with the $y$ terms sometimes resulted in an error: remember that $y$ means $y^1$, so $y^5 \times y = y^6$, not $y^5$.

The June 2025 examiner report found that "Candidates struggled to deal with the negative power appropriately in this question", and that common errors included multiplying the base number by the fractional power. A negative index never makes the answer negative — it makes it a reciprocal: $64^{-\frac{2}{3}} = \frac{1}{16}$, not $-16$ and not $64 \times -\frac{2}{3}$. Taking the root before the power keeps the arithmetic manageable ($16^3 = 4096$ is needless work).

The March 2026 examiner report noted that "Some candidates equated the powers as given in the question" — here that would be $x + 4 = x - 1$, which has no solution. Convert to a common base first, and multiply the whole index by $2$: $(3^2)^{x-1} = 3^{2x-2}$. Check: $3^{10} = 59\,049 = 9^5$. ✓

The March 2025 examiner report noted that the most successful candidates wrote each number as a product of two factors, one of which was a square number, as their starting point. The June 2025 report added that "Some common errors involved incorrect factorisation and failure to extract perfect square factors." The March 2026 report found candidates combining the numbers under the roots — but $\sqrt{75} + \sqrt{12}$ is not $\sqrt{87}$, and $\sqrt{50} - \sqrt{18}$ is not $\sqrt{32}$.

The June 2025 examiner report noted that "Incorrect responses generally did not attempt a method for rationalising denominators" — the first step is always to multiply top and bottom by the surd. 'Simplify' means the cancelling step is required: $\frac{10\sqrt{6}}{6}$ alone will not get the final mark. Check numerically: $\frac{10}{\sqrt6} \approx 4.08$ and $\frac{5\sqrt6}{3} \approx 4.08$. ✓

When candidates had to expand two surd brackets, the June 2025 examiner report observed that "The fully correct answer was only seen in a minority of cases". Write out all four terms before collecting. Part (a) shows why rationalising works: $(a+\sqrt b)(a-\sqrt b)$ always gives a rational result. In (b), $(3+\sqrt2)^2 \neq 9 + 2$ — the cross terms matter.

The March 2026 examiner report noted that, when squaring a surd length such as $4\sqrt{10}$, a common error was to square only the surd part. Here that slip gives $(2\sqrt{3})^2 = 2 \times 3 = 6$ — the $2$ must be squared too, giving $12$. In a 'show that', end by writing both values and saying they are equal; $\sqrt{18}$ left unsimplified in (a) loses the final mark.

In (c), $(5+\sqrt3)^2$ is not $25 + 3$: leaving out the $10\sqrt3$ cross terms happens to give the right total here only because they cancel, so the working still has to show them. In (d), multiplying by $5 - \sqrt{3}$ (the same sign) does not clear the surd from the denominator; and on rationalising a conjugate denominator, the June 2025 examiner report noted that "The main problem was multiplying the denominators correctly." Notice that parts (a) and (d) are linked: the denominator after rationalising is the area.

In (a) the half-sheet's sides must be matched long to long: $\frac{x}{1} = \frac{1}{\frac12 x}$, not $\frac{x}{1} = \frac{\frac12 x}{1}$. In (b) and (c), writing every surd as a power of $2$ turns the problem into adding and subtracting indices. As a check, a real A4 sheet is $29.7$ cm long.

The March 2025 examiner report noted that "Those who realised that they had to convert all the terms into the same base were often able to go on to gain all the available marks." The same report found a common incorrect equation in which the bracket was lost when multiplying an index — here, writing $3x - 2$ instead of $3(x - 2) = 3x - 6$. Keep brackets round every index until you expand them.

The June 2025 examiner report found that "some applied the power to just the algebraic part of the term", leaving the $16$ or the $125$ untouched. The March 2026 report described candidates who applied the fractional power to the number correctly and then did not deal with the negative sign — $25x^{4}$ or $\frac{25}{x^4}$ in (b). The reciprocal applies to both the number and the letter.

The June 2025 examiner report found that "Manipulating surds seemed to be challenging for many candidates." and that "many candidates did not realise that they had to equate the terms on the LHS and RHS to find c and k." Finish by stating $c$ and $k$ separately — an unfinished expansion does not answer the question.

The June 2025 examiner report advised that "it is useful to know the trigonometric ratios for the angles 0°, 30°, 45°, 60° and 90°, with the exception of tan90°." In a similar composite shape, the March 2025 report found it quite common for a calculated length to be used as if it were a different side when finding the perimeter. Label each length on the diagram as you find it, and leave out the shared side $BD$.