【PY100】Mathematics – (1) Laws of Indices & Scientific Notation

This is the first article in the Mathematics series, and we start with indices. The reason is simple: factorisation, quadratic equations, exponential functions and logarithms all rely on these few laws, right up to Form 6. When a Form 5 student gets an algebra question wrong, the cause is often an index rule from Form 2 that was never quite clear. So this is not a warm-up. It is the foundation for everything that follows.

First, the notation. $a^n$ means $n$ copies of $a$ multiplied together. For example, $2^3 = 2\times 2\times 2 = 8$. Note that an index applies only to the base right next to it: $-3^2$ is $-(3\times 3) = -9$, but $(-3)^2$ is $(-3)\times(-3) = 9$. Calculators and exam papers both read it this way, so whenever you see a minus sign, check for brackets.

Power: $a^n$ means $n$ copies of $a$ multiplied together, read "$a$ to the power $n$". 1. $a$ is the base. 2. $n$ is the index: it says how many times the base is multiplied.

Where the five laws come from

You do not need to memorise these laws. Count the bases and they follow. $a^3\times a^2$ is $(a\times a\times a)\times(a\times a)$. That is 5 copies of $a$, so it equals $a^5$ — the indices 3 and 2 are added. $(a^3)^2$ is $a^3\times a^3$: two groups of three, so $a^6$ — the indices 3 and 2 are multiplied. The other three laws come from counting in the same way. If you forget one, count it out again.

Laws of indices: for $a\neq 0$, $b\neq 0$ and integers $m$, $n$: 1. $a^m\times a^n = a^{m+n}$ — same base, multiply: add the indices. 2. $a^m\div a^n = a^{m-n}$ — same base, divide: subtract the indices. 3. $(a^m)^n = a^{mn}$ — a power of a power: multiply the indices. 4. $(ab)^n = a^nb^n$ — a power of a product: every factor is raised. 5. $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ — a power of a quotient: numerator and denominator are both raised.

There are three common mistakes. First, multiplying the indices when you should add them: $a^3\times a^2$ is $a^5$, not $a^6$. Second, forgetting the number inside the brackets: $(2x^3)^2 = 4x^6$, not $2x^6$, because the 2 is inside the brackets too. Third, combining indices when the bases are different: $2^3\times 3^2$ cannot be combined by the first law. Work each part out: $8\times 9 = 72$.

What zero and negative indices mean

By the second law, $a^3\div a^3 = a^{3-3} = a^0$. But any non-zero number divided by itself is 1, so $a^0$ must equal 1. In the same way, $a^2\div a^5 = a^{2-5} = a^{-3}$. Cancelling directly gives $\frac{a\times a}{a\times a\times a\times a\times a} = \frac{1}{a^3}$. The two answers must agree, so $a^{-3} = \frac{1}{a^3}$. Zero and negative indices are not new rules. They are what keeps the five laws true when an index is 0 or negative.

Zero and negative integral indices: for $a\neq 0$: 1. $a^0 = 1$. 2. $a^{-n} = \frac{1}{a^n}$. A negative index does not make the number negative. It means the power moves to the denominator (or up from it).

Two traps here. $2^{-3}$ is $\frac{1}{8}$, a positive number — not $-8$. And a negative index also applies only to the base next to it: $3a^{-2} = \frac{3}{a^2}$, and the 3 stays on top. Only $(3a)^{-2}$ equals $\frac{1}{9a^2}$. When a question says "express your answer with positive indices", this move is exactly what it is testing.

Example 1: simplify and use positive indices

This kind of question often opens DSE Paper 1: simplify $\frac{(2a^3b^{-1})^2}{4a^{-2}b^3}$ and express your answer with positive indices. One step at a time:

1. Expand the brackets. The power 2 outside applies to every factor inside, including the 2: $(2a^3b^{-1})^2 = 2^2a^{3\times 2}b^{-1\times 2} = 4a^6b^{-2}$
2. Divide the numbers: $\frac{4}{4} = 1$
3. Divide the powers of $a$ by subtracting the indices. This is the step most people get wrong — subtracting a negative means adding: $a^{6-(-2)} = a^{8}$
4. Do the same for $b$: $b^{-2-3} = b^{-5}$
5. Finally, move the negative index to the denominator: $a^8b^{-5} = \frac{a^8}{b^5}$

The answer is $\frac{a^8}{b^5}$. To check, substitute $a = 1$ and $b = 2$. The original expression gives $\frac{(2\times 1\times 2^{-1})^2}{4\times 1^{-2}\times 2^3} = \frac{1}{32}$, and the answer gives $\frac{1^8}{2^5} = \frac{1}{32}$. They match.

Scientific notation: writing very large and very small numbers

The most practical use of indices is writing numbers with too many zeros to count. The speed of light is about $300\,000\,000$ metres per second. A bacterium is about $0.000\,002$ metres long. Written like that, it is easy to miscount a place. Written with powers of 10, they become $3\times 10^8$ and $2\times 10^{-6}$, and the size is clear at a glance.

Scientific notation: a positive number written in the form $a\times 10^n$, where $1\le a<10$ and $n$ is an integer. 1. If the number is 10 or more, $n$ is positive. 2. If the number is at least 1 but less than 10, $n = 0$. 3. If the number is less than 1, $n$ is negative.

To convert, count how many places the decimal point moves. Move it left by some number of places and the index is that number, positive. Move it right and the index is that number, negative.

• $5\,030\,000 = 5.03\times 10^6$: the decimal point moves 6 places left.
• $0.000\,407 = 4.07\times 10^{-4}$: the decimal point moves 4 places right.

Example 2: multiplying in scientific notation

Find $(3.2\times 10^5)\times(4\times 10^{-8})$, giving your answer in scientific notation.

1. Separate the numbers from the powers of 10 and multiply each group: $(3.2\times 4)\times(10^5\times 10^{-8})$
2. The numbers: $3.2\times 4 = 12.8$
3. Same base, multiply: add the indices: $10^{5+(-8)} = 10^{-3}$
4. $12.8\times 10^{-3}$ is not yet in scientific notation, because 12.8 is more than 10. Write it as $1.28\times 10^1$ and use the first law once more: $1.28\times 10^1\times 10^{-3} = 1.28\times 10^{-2}$

The answer is $1.28\times 10^{-2}$. Many students stop at $12.8\times 10^{-3}$. The value is right, but it is not scientific notation, and the mark is lost. So every time, check that the front number satisfies $1\le a<10$. Check: $320\,000\times 0.000\,000\,04 = 0.0128$, which is $1.28\times 10^{-2}$.

The laws of indices simplify expressions; scientific notation writes numbers. They are one set of rules. Next time: change of subject — rearranging a formula to get the letter you want on its own. It uses the multiplying, dividing and powers from today constantly, so practise these five laws first.

Learn more: DSE Mathematics tutoring

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