Indices, Surds and Logarithms, Glossary

Knowing the vocabulary of Indices, Surds and Logarithms precisely is worth easy marks in SPM Add Math. Questions often turn on a single keyword, read it correctly and the method follows; misread it and even good algebra earns nothing.

Because the SPM paper is bilingual (BM/EN), each term below shows the English and Malay wording, so your child recognises it whichever way a question is phrased.

• Index, An index (or power) tells how many times a base number is multiplied by itself, as in $2^{5}=2\times2\times2\times2\times2$. In $a^{n}$, $a$ is the base and $n$ is the index. Indices can be positive, negative, zero, or fractional, and fractional indices represent roots. →
• Laws of Indices, The laws of indices are rules for combining powers with the same base, such as $a^{m}\times a^{n}=a^{m+n}$, $a^{m}\div a^{n}=a^{m-n}$, and $(a^{m})^{n}=a^{mn}$. They also give $a^{0}=1$ and $a^{-n}=\frac{1}{a^{n}}$. These laws simplify expressions and solve index equations. →
• Surd, A surd is a root that cannot be simplified to a rational number, such as $\sqrt{2}$ or $\sqrt{5}$, so its decimal never ends or repeats. Surds are exact forms of irrational numbers. We simplify them using $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$, for example $\sqrt{12}=2\sqrt{3}$. →
• Rationalising the Denominator, Rationalising the denominator removes a surd from the bottom of a fraction, giving a rational denominator. For $\frac{1}{\sqrt{a}}$ we multiply top and bottom by $\sqrt{a}$; for $\frac{1}{a+\sqrt{b}}$ we multiply by the conjugate $a-\sqrt{b}$. The value is unchanged, only the form is tidier. →
• Logarithm, A logarithm answers the question: to what power must a base be raised to give a number? It is the inverse of an index, so $a^{x}=n$ means $\log_{a} n=x$. For example $\log_{2} 8=3$ because $2^{3}=8$. The base must be positive and not equal to one. →
• Laws of Logarithms, The laws of logarithms turn products, quotients, and powers into simpler operations: $\log_a xy=\log_a x+\log_a y$, $\log_a\frac{x}{y}=\log_a x-\log_a y$, and $\log_a x^{n}=n\log_a x$. They let us expand, combine, or solve logarithmic equations, and they mirror the laws of indices. →
• Change of Base Formula, The change of base formula rewrites a logarithm using a different, more convenient base: $\log_a b=\frac{\log_c b}{\log_c a}$. It is essential when a calculator only has base 10 or base $e$, and it also helps simplify expressions like $\log_a b\times\log_b a=1$. →
• Common Logarithm, A common logarithm is a logarithm with base ten, written simply as $\log x$ or $\lg x$. Because our number system is base ten, common logarithms are handy for very large or very small numbers and appear on most calculators. For example $\log 1000=3$. →

Do not just memorise definitions, practise using each term inside a full solution, the way a marker expects to see it. In the Indices, Surds and Logarithms chapter especially, stating the right definition or condition in your working can itself earn a method mark.

Our teachers check that a student can both recall a term and deploy it fluently under exam conditions.

Vocabulary sticks best when it is tied to doing rather than reading. Instead of learning the Indices, Surds and Logarithms terms as a flat list, meet each one inside a worked question and say the step aloud in words as you write it, a term you can use in a sentence is a term you understand.

It also helps to link related terms together rather than in isolation, since Indices, Surds and Logarithms questions usually combine several ideas at once. Because the SPM paper is bilingual, glance once at the Malay and English wording of each term side by side, so the language a question is set in never throws you.

If a term keeps feeling slippery, that is usually a sign the underlying idea needs another look, not the word itself, and that is exactly the kind of gap a one-to-one lesson closes quickly.