Year 11 Methods (Unit 1 & 2) Exponential Functions And Logarithms

Standard Form

Q: What is standard form?

Standard form, also called scientific notation, writes a number as a coefficient times a power of ten. The coefficient is at least one and less than ten, and the power is a whole number.

Q: How do you write a small number in standard form?

Move the decimal point to the right until one non-zero digit is in front of it, and make the power negative. For example zero point zero zero zero three zero eight equals three point zero eight times ten to the minus four.

Q: How do you multiply numbers in standard form?

Multiply the coefficients and add the powers of ten, then renormalise so the coefficient is at least one and less than ten. Six times ten to the seven, times three times ten to the minus three, gives one point eight times ten to the five.

Q: How do you add numbers in standard form?

First make the powers of ten equal, then add the coefficients and renormalise if needed. Four point five times ten to the six plus zero point seven times ten to the six equals five point two times ten to the six.

Q: What are significant figures?

Significant figures are the meaningful digits of a number, counted from the first non-zero digit. They tell you how precisely a value has been rounded.

20 practice questions 2 video lessons Theory + worked examples

Learn standard form, or scientific notation, for Queensland Year 11 Mathematical Methods (QCAA). It writes numbers as a value between one and ten multiplied by a power of ten, keeping large and small numbers tidy.

You will learn to write numbers in standard form, round to a number of significant figures, and multiply, divide, add and subtract in this notation — handy for science and powers of ten.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)

Standard Form - Video - Standard Form Watch
Standard Form - Video - Scientific Notation and Significant Figures Watch

Theory

In Year 11 Mathematical Methods (QCAA, Unit 2), standard form (scientific notation) writes any number as \(a\times 10^{n}\), where \(1\le a<10\) and \(n\) is a whole number. This page shows how to write numbers in and out of standard form, round to significant figures, and multiply, divide, add and subtract in standard form.

Standard form (also called scientific notation) writes a number as a coefficient times a power of ten, \(a\times 10^{n}\), where the coefficient satisfies \(1\le a<10\) and \(n\) is an integer. A positive \(n\) makes a large number; a negative \(n\) makes a small number.

To build the coefficient, move the decimal point until exactly one non-zero digit sits in front of it; the number of places moved is the power of ten. Significant figures count the meaningful digits, starting from the first non-zero digit, which lets you round large or small numbers sensibly.

Coefficient rule: the front number must be at least \(1\) and less than \(10\). If a calculation gives \(18\times10^{4}\), renormalise it to \(1.8\times10^{5}\).

Writing \(47\,300=4.73\times10^{4}\): the decimal point moves \(4\) places, so \(n=4\).

The power of ten fixes the size: \(10^{-3}\) is thousandths, \(10^{6}\) is millions.

A number in standard form, with \(1\le a<10\) and integer \(n\):

\[a\times 10^{n}\]

a \times 10^{n}

Multiplying and dividing use the index laws on the powers of ten:

\[(a\times10^{m})\times(b\times10^{n})=(a\times b)\times10^{m+n}\]

(a \times 10^{m}) (b \times 10^{n}) = a b \times 10^{m + n}

Add and subtract: first make the powers of ten equal, then add or subtract the coefficients and renormalise so \(1\le a<10\).

How to write a number in standard form

Place the point: move the decimal point so one non-zero digit is in front of it, giving the coefficient \(a\) with \(1\le a<10\).
Count the moves: the number of places moved is the power \(n\) — positive for a large number (point moved left), negative for a small number (point moved right).
Write it: record the answer as \(a\times 10^{n}\), and renormalise if any calculation pushes \(a\) outside \([1,10)\).

Example 1 — Writing a large number in standard form

Write \(47\,300\) in standard form.

Solution

Put one non-zero digit before the point — the coefficient is \(4.73\):

\(a\)

\(=\)

\(4.73\)

Count the places the point moved (left) for the power:

\(47\,300\)

\(=\)

\(4.73\times 10^{4}\)

\(47\,300=4.73\times10^{4}\).

4.73 \times 10^{4}

Example 2 — Back to an ordinary number

Write \(3.08\times10^{-4}\) as an ordinary decimal number.

Solution

The power \(-4\) means move the decimal point \(4\) places left:

\(3.08\times10^{-4}\)

\(=\)

\(0.000308\)

Check: three zeros then \(308\) sits four places below the units.

\(3.08\times10^{-4}=0.000308\).

0.000308

Example 3 — Multiplying in standard form

Evaluate \((6\times10^{7})\times(3\times10^{-3})\), giving your answer in standard form.

Solution

Multiply the coefficients, add the powers of ten:

\((6\times 3)\times 10^{7+(-3)}\)

\(=\)

\(18\times 10^{4}\)

Renormalise so \(1\le a<10\) — \(18=1.8\times10\):

	\(=\)	\(1.8\times10\times10^{4}\)
	\(=\)	\(1.8\times10^{5}\)

\((6\times10^{7})\times(3\times10^{-3})=1.8\times10^{5}\).

1.8 \times 10^{5}

Example 4 — Adding in standard form

Evaluate \((4.5\times10^{6})+(7\times10^{5})\), giving your answer in standard form.

Solution

Make the powers of ten equal — rewrite \(7\times10^{5}\) as \(0.7\times10^{6}\):

\(7\times10^{5}\)

\(=\)

\(0.7\times10^{6}\)

Add the coefficients over the common power:

\((4.5+0.7)\times10^{6}\)

\(=\)

\(5.2\times10^{6}\)

\((4.5\times10^{6})+(7\times10^{5})=5.2\times10^{6}\).

5.2 \times 10^{6}

Common pitfalls

A coefficient outside \([1,10)\). \(18\times10^{4}\) is not in standard form; renormalise to \(1.8\times10^{5}\). The front number must be at least \(1\) and less than \(10\).

Wrong sign on the power. Large numbers take a positive power, small numbers (below \(1\)) take a negative power, so \(0.000308=3.08\times10^{-4}\), not \(10^{4}\).

Adding the powers when adding numbers. The index laws only add powers for multiplication. To add or subtract, first equalise the powers of ten.

Frequently asked questions

What is standard form?

Standard form (scientific notation) writes a number as \(a\times10^{n}\), where the coefficient satisfies \(1\le a<10\) and \(n\) is a whole number.

How do you write a small number in standard form?

Move the decimal point right until one non-zero digit is in front of it, and make the power negative. For example \(0.000308=3.08\times10^{-4}\).

How do you multiply numbers in standard form?

Multiply the coefficients and add the powers of ten, then renormalise so \(1\le a<10\). For example \((6\times10^{7})(3\times10^{-3})=1.8\times10^{5}\).

How do you add numbers in standard form?

First make the powers of ten equal, then add the coefficients and renormalise if needed, as in \(4.5\times10^{6}+0.7\times10^{6}=5.2\times10^{6}\).

What are significant figures?

Significant figures are the meaningful digits of a number counting from the first non-zero digit; they tell you how precisely a value is rounded.

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