Standard Form GCSE Foundation: Indices Made Clear

Standard form GCSE foundation: start with the meaning

A small raised number can make an ordinary calculation feel unfamiliar. If you are resitting GCSE Maths, that feeling may arrive before you have even read the question. Standard form GCSE foundation questions become more manageable when you connect those powers to something familiar: multiplying and dividing by ten.

Standard form writes a positive number as $a\times 10^n$, where $1\le a<10$ and $n$ is an integer. Indices are powers, and index laws help you calculate with them. Together, these ideas let you handle very large and very small numbers without writing every zero.

You can begin with MathsGenie's standard form revision lessons, then use the examples here to practise each decision.

Your revision checklist

Aim to recognise and complete these tasks:

• Explain what a power means.
• Convert ordinary numbers into standard form and back.
• Multiply and divide numbers in standard form.
• Add and subtract by expressing quantities consistently.
• Apply index laws when the bases match.
• Use a calculator and check whether its answer is sensible.

These topics belong in foundation revision. You do not need to master every possible indices question before practising the foundation skills.

Understand powers before learning the rules

An index tells you how many times a base appears as a factor. For example:

$$3^4=3\times 3\times 3\times 3=81$$

The base is $3$ and the index is $4$. It does not mean $3\times 4$.

Powers of ten connect indices to place value:

$$10^3=1000,10^0=1,10^{-3}=0.001$$

A negative power of ten represents a reciprocal. It does not make the number negative. In particular, $10^{-3}=\frac{1}{1000}$.

If powers GCSE questions still feel uncertain, work through MathsGenie's indices revision resources before combining them with standard form.

Convert numbers into standard form

Standard form has two parts: a coefficient between $1$ and $10$, excluding $10$, and a power of ten that preserves the original value.

A large number without a calculator

Write $630000$ in standard form.

First choose $6.3$ as the coefficient. Then ask what multiplies it back to the original number:

$$6.3\times 100000=630000$$

Therefore:

$$630000=6.3\times 10^5$$

Thinking about the multiplier makes the sign of the power easier to justify than memorising a direction rule.

A small number without a calculator

Write $0.00072$ in standard form.

The coefficient is $7.2$. To recover the original value, divide it by $10000$:

$$0.00072=7.2\times 10^{-4}$$

For positive numbers smaller than $1$, the standard form exponent is negative.

Convert back and check

To turn standard form into an ordinary number, apply the power of ten:

$$4.8\times 10^3=4800$$ $$4.8\times 10^{-3}=0.0048$$

On a calculator, enter the coefficient, multiplication sign, and the power using brackets where needed. For the second example, enter $4.8\times 10^{(-3)}$. The display gives a useful check, but you should still predict whether the answer will be large or small.

A hand-drawn stick-figure comic about very large and very small numbers sharing the same helpful power-of-ten key

Use index laws GCSE questions rely on

Index laws describe patterns in repeated multiplication. The matching base is essential.

Multiply powers with the same base

Add the indices:

$$a^m\times a^n=a^{m+n}$$

For example:

$$2^3\times 2^4=2^7=128$$

Without a calculator, expanding the factors explains why there are seven copies of $2$. With a calculator, check that $8\times 16=128$.

Divide powers with the same base

Subtract the indices, provided the base is non-zero:

$$a^m÷a^n=a^{m-n}$$

For example:

$$5^6÷5^2=5^4=625$$

You can simplify first without a calculator, then evaluate the remaining power if needed.

Raise a power to another power

Multiply the indices:

$$(a^m{)}^n=a^{mn}$$

For example:

$$(3^2{)}^3=3^6=729$$

Also remember that $a^0=1$ for $a\ne 0$. MathsGenie's indices revision guide provides another place to revisit these relationships.

Calculate with standard form

The useful habit is to handle the coefficients and powers separately, then check that the result satisfies the standard form definition.

Non-calculator multiplication

Calculate:

$$(3\times 10^4)(4\times 10^2)$$

Multiply the coefficients and add the powers:

$$(3\times 4)\times 10^{4+2}=12\times 10^6$$

The value is correct, but the coefficient is too large. Rewrite it:

$$12\times 10^6=1.2\times 10^7$$

Calculator division

Calculate:

$$\frac{7.2\times 10^8}{3\times 10^3}$$

Enter the numerator and denominator separately in a fraction template, or put brackets around each complete number. The calculation is:

$$\frac{7.2}{3}\times 10^{8-3}=2.4\times 10^5$$

A calculator may display $240000$ or use scientific notation, depending on its settings. Both represent the same value, but give standard form when requested.

Addition and subtraction

You cannot use the multiplication rule to add powers. First express both quantities using the same power of ten:

$$3.2\times 10^5+4\times 10^4=3.2\times 10^5+0.4\times 10^5=3.6\times 10^5$$

For subtraction:

$$6.5\times 10^4-2\times 10^4=4.5\times 10^4$$

With a calculator, entering the complete quantities also works. Without one, matching powers keeps the arithmetic small. Practise both approaches using the standard form question booklet.

Common mistakes and how to catch them

Leaving the coefficient outside the allowed range. An answer such as $25\times 10^3$ needs rewriting as $2.5\times 10^4$.

Treating a negative exponent as a negative number. The value $10^{-2}$ is $0.01$, not $-100$.

Applying index laws to different bases. You cannot combine $2^3\times 5^4$ by simply adding the indices.

Using multiplication rules for addition. The expression $10^2+10^3$ equals $1100$, not $10^5$.

Losing brackets on the calculator. Keep each standard form quantity together, especially when dividing. Estimate the size of the answer before accepting the display.

Turn revision into a repeatable routine

For a resit, a useful session has a clear purpose: identify one mistake, correct the reasoning, then answer a fresh question independently.

Start with conversions. Next practise index laws, followed by mixed calculations. Use MathsGenie's mini tests for a short check, then select your exam board through the GCSE Maths revision hub for past papers and mark schemes. Review video solutions where available, and attempt the question again without looking.

Predicted papers can provide additional practice, but they cannot guarantee exam content. Use them alongside topic questions and past papers. Put your next practice session into the revision planner.

Confidence grows when the notation starts to mean something. Open MathsGenie's free revision resources, complete a short set, and check every answer. Your next step is simply to understand one more question than you did yesterday.