Worked example · Easy
Write a large number in standard form
Write \(583000\) in standard form.
Move the decimal point 5 places left:
\[583000=5.83\times10^5\]Answer: \(\boxed{5.83\times10^5}\)
Chapter 8
Standard form
Write very large and very small numbers efficiently using powers of 10, then calculate accurately with standard-form values.
By the end of this chapter, you should be able to
- write numbers in the form \(A\times10^n\), where \(1\le A<10\) and \(n\) is an integer;
- convert very large and very small ordinary numbers into standard form;
- convert standard-form values back to ordinary numbers;
- multiply and divide standard-form values using index laws;
- add and subtract standard-form values accurately;
- normalise a calculation so the final coefficient is between 1 and 10;
- enter and interpret standard-form values correctly on a calculator.
1
Subtopic 1
Standard form
Standard form is used to make very large and very small numbers easier to write and calculate with.
The required form
\[\boxed{A\times10^n}\]where \(1\le A<10\) and \(n\) is a positive or negative integer.
Large number4 720 000→\(4.72\times10^6\)Decimal point moves 6 places left.
Small number0.000 083→\(8.3\times10^{-5}\)Decimal point moves 5 places right, so the index is negative.
Ordinary number → standard form
- Move the decimal point until the first factor is between 1 and 10.
- Count how many places the decimal point moved.
- Large original number → positive power of 10.
- Small original number between 0 and 1 → negative power of 10.
Exam method
The first factor must satisfy \(1\le A<10\); rewrite anything like \(34 imes10^5\) as \(3.4 imes10^6\).
Important: In standard form, the coefficient may equal 1 but must be less than 10. The exponent can be zero, positive or negative.
Worked example · Easy
Write a small number in standard form
Write \(0.0000472\) in standard form.
Move the decimal point 5 places right to obtain 4.72. Because the original number is smaller than 1, the index is negative.
\[\boxed{4.72\times10^{-5}}\]Worked example · Medium
Convert back to an ordinary number
Write \(6.04\times10^{-4}\) as an ordinary number.
The power is \(-4\), so move the decimal point 4 places left:
\[6.04\times10^{-4}=\boxed{0.000604}\]Worked example · Medium
A value written in millions
Write 38.6 million in standard form.
38.6 million = 38 600 000.
\[38\,600\,000=\boxed{3.86\times10^7}\]Worked example · Exam-style extension
Convert values both ways
Write \(0.000000708\) in standard form, then give \(8.15\times10^6\) as an ordinary number.
Move the point seven places to the right to get 7.08.
\[0.000000708=\boxed{7.08\times10^{-7}}\]
For the second number move the point six places to the right.
\[8.15\times10^6=\boxed{8\,150\,000}\]
Common mistakes
- \(47\times10^5\) is not in standard form because 47 is not between 1 and 10.
- A negative index does not mean the number is negative; it indicates a small positive number when the coefficient is positive.
- Do not count the digits. Count the number of places the decimal point moves.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Write \(92\,000\) in standard form.
Show Answer
Move the point four places left.
\[\boxed{9.2\times10^4}\]
2Apply the method[4 marks]
Write \(0.0000432\) in standard form.
Show Answer
Move the point five places right; use a negative index.
\[\boxed{4.32\times10^{-5}}\]
3Exam-style challenge[5 marks]
Write \(3.06\times10^{-4}\) as an ordinary decimal, then compare it with \(0.00031\).
Show Answer
\(3.06\times10^{-4}=0.000306\).
Since \(0.000306<0.000310\),
\[\boxed{3.06\times10^{-4}<0.00031}\]
2
Subtopic 2
Calculating with numbers in standard form
For multiplication and division, work with the number parts and powers of 10 separately. For addition and subtraction without a calculator, the source method changes the values to ordinary numbers first.
Multiply\[(a\times10^m)(b\times10^n)=ab\times10^{m+n}\]
Divide\[\frac{a\times10^m}{b\times10^n}=\frac ab\times10^{m-n}\]
Exam method
For products or quotients, use index laws; for sums/differences align place values (or powers of 10), then normalise.
Worked example · Easy
Multiply standard-form values
Work out \((3\times10^4)(2\times10^3)\).
\[3\times2=6\]\[10^4\times10^3=10^7\]\[\boxed{6\times10^7}\]
Worked example · Medium
Normalise after multiplication
Work out \((6\times10^5)(4\times10^3)\). Give the answer in standard form.
\[(6\times4)\times10^{5+3}=24\times10^8\]
24 is too large for the coefficient, so rewrite:
\[24\times10^8=\boxed{2.4\times10^9}\]Worked example · Medium
Divide standard-form values
Work out \((8.4\times10^7)\div(2.1\times10^3)\).
\[\frac{8.4}{2.1}=4\]\[10^{7-3}=10^4\]\[\boxed{4\times10^4}\]
Worked example · Exam-style
Normalise after division
Work out \((3.6\times10^5)\div(6\times10^{-2})\). Give the answer in standard form.
\[\frac{3.6}{6}\times10^{5-(-2)}=0.6\times10^7\]
The coefficient 0.6 is too small for standard form:
\[0.6\times10^7=\boxed{6\times10^6}\]Addition and subtraction
For the method used in this chapter, convert each value to an ordinary number, add or subtract carefully, then convert the result back to standard form.
Worked example · Easy
Add standard-form values
Work out \((5\times10^3)+(2.4\times10^4)\).
\[5000+24000=29000\]\[\boxed{2.9\times10^4}\]
Worked example · Medium
Subtract small standard-form values
Work out \((6.2\times10^{-3})-(7.5\times10^{-4})\).
\[0.00620-0.00075=0.00545\]\[\boxed{5.45\times10^{-3}}\]
Worked example · Exam-style
Square a standard-form value
Work out \((4.5\times10^3)^2\). Give the answer in standard form.
\[(4.5)^2\times(10^3)^2=20.25\times10^6\]
Normalise:
\[20.25\times10^6=\boxed{2.025\times10^7}\]Worked example · Exam-style extension
Operations and normalisation
Calculate \(\frac{(6\times10^{-4})(3\times10^7)}{9\times10^2}\) in standard form.
Combine numerical parts: \(6\times3\div9=2\).
Combine powers: \(10^{-4+7-2}=10^1\).
\[\boxed{2\times10^1}\]
Calculator check
Scientific calculators usually have an EXP, EE or \(\times10^x\) style key for standard-form entry. Learn which key your calculator uses and read the displayed exponent carefully.
Final check after every calculation
Ask: “Is the coefficient at least 1 and less than 10?” If not, move the decimal point and adjust the power of 10.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Calculate \((4\times10^3)(5\times10^2)\) in standard form.
Show Answer
\(4\times5=20\); powers give \(10^5\).
\[20\times10^5=\boxed{2\times10^6}\]
2Apply the method[4 marks]
Work out \((8\times10^{-3})\div(2\times10^2)\).
Show Answer
\(8\div2=4\), power \(-3-2=-5\).
\[\boxed{4\times10^{-5}}\]
3Exam-style challenge[5 marks]
A laboratory combines \(3.5\times10^{-4}\text{ L}\) and \(7.2\times10^{-5}\text{ L}\). Give the total volume in standard form.
Show Answer
Write both with the same exponent:
\[3.5\times10^{-4}+0.72\times10^{-4}=4.22\times10^{-4}\]
\[\boxed{4.22\times10^{-4}\text{ L}}\]
Chapter 8 practice
Begin with conversions, then move to calculations and mixed exam-style problems. Keep each answer hidden until you have attempted the question.
A. Build confidence
1Large numbers[3 marks]
Write in standard form: a) 7200 b) 684000 c) 39 500 000.
Show Answer
a) \(7200=\boxed{7.2\times10^3}\)
b) \(684000=\boxed{6.84\times10^5}\)
c) \(39\,500\,000=\boxed{3.95\times10^7}\)
2Small numbers[3 marks]
Write in standard form: a) 0.0042 b) 0.000071 c) 0.000000608.
Show Answer
a) \(\boxed{4.2\times10^{-3}}\)
b) \(\boxed{7.1\times10^{-5}}\)
c) \(\boxed{6.08\times10^{-7}}\)
3Back to ordinary numbers[4 marks]
Write as ordinary numbers: a) \(3.7\times10^4\) b) \(9.02\times10^6\) c) \(5.6\times10^{-3}\) d) \(2.41\times10^{-5}\).
Show Answer
a) \(\boxed{37000}\)
b) \(\boxed{9\,020\,000}\)
c) \(\boxed{0.0056}\)
d) \(\boxed{0.0000241}\)
4Is it standard form?[4 marks]
Rewrite each correctly in standard form: a) \(32\times10^5\) b) \(0.7\times10^8\) c) \(15.4\times10^{-6}\) d) \(0.09\times10^{-3}\).
Show Answer
a) \(32\times10^5=\boxed{3.2\times10^6}\)
b) \(0.7\times10^8=\boxed{7\times10^7}\)
c) \(15.4\times10^{-6}=\boxed{1.54\times10^{-5}}\)
d) \(0.09\times10^{-3}=\boxed{9\times10^{-5}}\)
B. Calculating
5Multiply[3 marks]
Work out \((4\times10^6)(3\times10^2)\). Give the answer in standard form.
Show Answer
\[4\times3\times10^{6+2}=12\times10^8=\boxed{1.2\times10^9}\]
6Multiply with negative indices[4 marks]
Work out \((7\times10^{-4})(5\times10^{-3})\).
Show Answer
\[7\times5\times10^{-7}=35\times10^{-7}=\boxed{3.5\times10^{-6}}\]
7Divide[4 marks]
Work out \((9\times10^8)\div(3\times10^4)\).
Show Answer
\[\frac93\times10^{8-4}=\boxed{3\times10^4}\]
8Division and normalisation[4 marks]
Work out \((2.4\times10^3)\div(8\times10^{-2})\).
Show Answer
\[\frac{2.4}{8}\times10^{3-(-2)}=0.3\times10^5=\boxed{3\times10^4}\]
9Add[4 marks]
Work out \((3.2\times10^5)+(4.7\times10^4)\).
Show Answer
\[320000+47000=367000\]\[\boxed{3.67\times10^5}\]
10Subtract[4 marks]
Work out \((8.1\times10^{-3})-(6.4\times10^{-4})\).
Show Answer
\[0.00810-0.00064=0.00746\]\[\boxed{7.46\times10^{-3}}\]
C. Exam-style challenge
11Population[3 marks]
A country has a population of 52.8 million. Write this population in standard form.
Show Answer
52.8 million = 52 800 000.
\[52\,800\,000=\boxed{5.28\times10^7}\]12Very small measurement[4 marks]
A thickness is \(0.0000000345\) m. Write it in standard form.
Show Answer
Move the decimal point 8 places right:
\[\boxed{3.45\times10^{-8}\text{ m}}\]13Area from standard form[5 marks]
A square has side length \(3.2\times10^4\) cm. Find its area in \(\text{cm}^2\), giving the answer in standard form.
Show Answer
\[A=(3.2\times10^4)^2\]\[=10.24\times10^8\]\[=\boxed{1.024\times10^9\text{ cm}^2}\]
14Combined calculation[5 marks]
Work out \(\dfrac{(6\times10^7)(2.5\times10^{-3})}{3\times10^2}\). Give your answer in standard form.
Show Answer
Numerator:
\[(6\times2.5)\times10^{7-3}=15\times10^4\]Now divide:
\[\frac{15\times10^4}{3\times10^2}=5\times10^2\]\(\boxed{5\times10^2}\)
15Spot and correct an error[4 marks]
A student writes \((8\times10^5)(4\times10^3)=32\times10^8\) and stops. Explain what is incomplete and give the final answer in standard form.
Show Answer
The multiplication is numerically correct, but 32 is not an allowed standard-form coefficient.
\[32\times10^8=\boxed{3.2\times10^9}\]16Difference of small values[5 marks]
Find \((7.4\times10^{-5})-(2.85\times10^{-5})\), giving the answer in standard form.
Show Answer
\[0.000074-0.0000285=0.0000455\]\[\boxed{4.55\times10^{-5}}\]
Chapter 8 high-grade checklist
Correct form
I check that the coefficient satisfies \(1\le A<10\).
Convert both ways
I can move accurately between ordinary numbers and powers of 10.
Signs of indices
I use positive powers for large values and negative powers for small positive values below 1.
Multiply/divide
I calculate coefficients and powers separately, then normalise.
Add/subtract
I can safely convert to ordinary numbers before combining values.
Final check
I never leave answers such as \(24\times10^7\) or \(0.6\times10^5\) as standard form.
Illustrations are taken only from actual Chapter 8 diagrams where present; explanations and practice questions are newly written. Text-only book pages are not reproduced as images.