Module 7 of 46Lesson 1 of 10 · AA 1.1 — Operations with numbers in standard form

Course map · Topic 1: Number and Algebra

1.1 Operations with numbers in standard form

SL & HL11 worked questions

Standard form is a short way to write very big and very small numbers, like the mass of an atom or the value of a diamond mine. On Paper 1 you must multiply, divide, add and subtract them by hand; on Paper 2 the GDC does the arithmetic, but you still write the answer correctly.

✏️ Start with the questions. Try each one on paper, then open the solution and compare.

Stuck? Open the notes ↓

✏️ Practice questions

Try each question on paper first. Then open the solution and compare every step.

Question 1●○○ Warm-up✍️ By hand · Paper 1 · no calculator[2 marks]

Write each number in the form

a\times10^{k}

, where

1\le a\lt 10

and

k\in\mathbb{Z}

(a) $52\,300\,000$ [1]
(b) $0.000\,407$ [1]

Show the full step-by-step solution

Step 1🧠 ThinkWhere must the decimal point go?

Move the decimal point until there is exactly one non-zero digit in front of it. Count how many places you moved it — that count is the power of 10.

Part (a)

Step 1✍️ By handA big number → positive power

52\,300\,000

: the point moves 7 places to the left to give 5.23.

52\,300\,000 = 5.23\times10^{7}

Part (b)

Step 1✍️ By handA small number → negative power

0.000\,407

: the point moves 4 places to the right to give 4.07.

0.000\,407 = 4.07\times10^{-4}

✅ Check it

4.07\times10^{-4} = 4.07\div10\,000 = 0.000\,407

✔

Final answer (a)

5.23\times10^{7}

(b)

4.07\times10^{-4}

Question 2●○○ Warm-up✍️ By hand · Paper 1 · no calculator[2 marks]

Calculate

(3\times10^{5})\times(4\times10^{-2})

, giving your answer in the form

a\times10^{k}

, where

1\le a\lt 10

and

k\in\mathbb{Z}

Show the full step-by-step solution

Step 1🧠 ThinkMultiply the numbers, add the powers

Put the front numbers together and the powers of 10 together:

3\times4

and

10^{5}\times10^{-2}

Step 2✍️ By handWork it out

= (3\times4)\times10^{5+(-2)}

= 12\times10^{3}

= 1.2\times10^{4}

💡 Remember

12\times10^{3}

is the right value but not standard form.

12 = 1.2\times10

, so one extra power of 10 goes onto the 3.

Final answer

1.2\times10^{4}

⚠️ Common mistake Stopping at

12\times10^{3}

loses the final A1.

Question 3●○○ Warm-up✍️ By hand · Paper 1 · no calculator[2 marks]

Calculate

\dfrac{8.4\times10^{6}}{2.1\times10^{-3}}

, giving your answer in standard form.

Show the full step-by-step solution

Step 1🧠 ThinkDivide the numbers, subtract the powers

8.4\div2.1

is easy:

84\div21 = 4

. For the powers:

6 - (-3)

. Subtracting a negative is the same as adding.

Step 2✍️ By handWork it out

= \dfrac{8.4}{2.1}\times10^{6-(-3)}

= 4\times10^{9}

Final answer

4\times10^{9}

⚠️ Common mistake Writing

10^{6-3} = 10^{3}

. The power on the bottom is

-3

, so

6-(-3) = 9

Question 4●●○ Standard✍️ By hand · Paper 1 · no calculator[7 marks]

Let

a = 6\times10^{4}

and

b = 5\times10^{3}

. Find each of the following, giving your answers in standard form.

(a) $a+b$ [2]
(b) $a-b$ [1]
(c) $ab$ [2]
(d) $\dfrac{a}{b}$ [2]

Show the full step-by-step solution

Step 1🧠 ThinkAdding is different from multiplying

To add or subtract, both numbers must use the same power of 10 — like adding money only when both amounts are in pula. Write

a = 60\times10^{3}

so that it matches

b

Part (a)

Step 1✍️ By handSame power, then add

a+b = 60\times10^{3} + 5\times10^{3}

= 65\times10^{3}

= 6.5\times10^{4}

Part (b)

Step 1✍️ By handSame idea, subtract

a-b = 55\times10^{3} = 5.5\times10^{4}

Part (c)

Step 1✍️ By handMultiply: add the powers

ab = (6\times5)\times10^{4+3}

= 30\times10^{7}

= 3\times10^{8}

Part (d)

Step 1✍️ By handDivide: subtract the powers

\dfrac{a}{b} = \dfrac{6}{5}\times10^{4-3}

= 1.2\times10^{1}

(that is, 12)A1

Final answer (a)

6.5\times10^{4}

(b)

5.5\times10^{4}

(c)

3\times10^{8}

(d)

1.2\times10^{1}

Question 5●●○ Standard✍️ By hand · Paper 1 · no calculator[4 marks]

Find the exact value of each expression, giving your answer in standard form.

(a) $\sqrt{3.6\times10^{7}}$ [2]
(b) $\left(1.5\times10^{-4}\right)^{2}$ [2]

Show the full step-by-step solution

Step 1🧠 ThinkMake the power of 10 easy to work with

A square root halves the power, so the power must be even. Rewrite

3.6\times10^{7}

36\times10^{6}

— same number, even power, and 36 is a perfect square.

Part (a)

Step 1✍️ By handRewrite, then take the square root

3.6\times10^{7} = 36\times10^{6}

\sqrt{36\times10^{6}} = \sqrt{36}\times\sqrt{10^{6}} = 6\times10^{3}

= 6\times10^{3}

Part (b)

Step 1✍️ By handSquare both parts

\left(1.5\times10^{-4}\right)^{2} = 1.5^{2}\times10^{-8}

= 2.25\times10^{-8}

💡 Remember

\left(10^{-4}\right)^{2} = 10^{-4\times2} = 10^{-8}

: a power of a power multiplies.

Final answer (a)

6\times10^{3}

(b)

2.25\times10^{-8}

⚠️ Common mistake Writing

\sqrt{3.6}\times10^{3.5}

. Always make the power even first.

Question 6●●○ Standard✍️ By hand · Paper 1 · no calculator[5 marks]

Let

p = 4\times10^{2n}

and

q = 8\times10^{n-1}

, where

n\in\mathbb{Z}

. Giving each answer in the form

a\times10^{k}

, where

1\le a\lt 10

and

k

is an expression in

n

, find

(a) $pq$ [3]
(b) $\dfrac{p}{q}$ [2]

Show the full step-by-step solution

Step 1🧠 ThinkThe rules work with letters too

Multiply the front numbers and add the powers

2n

and

n-1

. Then tidy up the front number exactly as you would with ordinary numbers.

Part (a)

Step 1✍️ By handMultiply

pq = 32\times10^{2n+n-1} = 32\times10^{3n-1}

= 3.2\times10^{1}\times10^{3n-1}

= 3.2\times10^{3n}

Part (b)

Step 1✍️ By handDivide

\dfrac{p}{q} = \dfrac{4}{8}\times10^{2n-(n-1)} = 0.5\times10^{n+1}

= 5\times10^{-1}\times10^{n+1} = 5\times10^{n}

✅ Check it Try

n = 1

p = 400,\ q = 8

. Then

pq = 3200 = 3.2\times10^{3}

✔ and

\dfrac{p}{q} = 50 = 5\times10^{1}

✔

Final answer (a)

3.2\times10^{3n}

(b)

5\times10^{n}

⚠️ Common mistake Writing

2n-n-1

without the bracket in (b). It must be

2n-(n-1) = n+1

Question 7●●○ Standard🖩 GDC · Paper 2[3 marks]

One mole of water has a mass of

18.02

grams and contains

6.022\times10^{23}

molecules. Find the mass, in grams, of one water molecule. Give your answer in the form

a\times10^{k}

, where

1\le a\lt 10

, to 3 significant figures.

Show the full step-by-step solution

Step 1🧠 ThinkShare the mass equally between the molecules

6.022\times10^{23}

molecules weigh 18.02 g together, one molecule weighs

18.02\div(6.022\times10^{23})

Step 2🖩 On the GDCUse the EE key for the power of 10

18.02›÷💻 /›6.022›EE💻 click EE on keypad›23›enter💻 Enter

With Exponential Format set to Scientific the answer appears in standard form straight away.

2.99236E−23 means 2.99236 × 10⁻²³

Step 3✍️ By handWrite it properly and round

\dfrac{18.02}{6.022\times10^{23}}

= 2.99\times10^{-23}

gA1A1

Final answer

2.99\times10^{-23}

⚠️ Common mistake Writing 2.99E−23. In the exam this is not accepted as standard form.

Question 8●●○ Standard🖩 GDC · Paper 2[4 marks]

The mass of the Earth is

5.97\times10^{24}

kg and the mass of the Moon is

7.35\times10^{22}

kg.

(a) Find how many times heavier the Earth is than the Moon. Give your answer to 3 significant figures.[2]
(b) Find the total mass of the Earth and the Moon, in standard form, to 3 significant figures.[2]

Show the full step-by-step solution

Step 1🧠 Think“How many times” means divide

To compare two amounts by “how many times”, divide the big one by the small one. For the total, add — and remember the powers are different, so let the GDC line them up.

Step 2🖩 On the GDCBoth calculations on one page

5.97›EE💻 click EE on keypad›24›÷💻 /›7.35›EE💻 click EE on keypad›22›enter💻 Enter›5.97›EE💻 click EE on keypad›24›+›7.35›EE💻 click EE on keypad›22›enter💻 Enter

81.2245 times; total 6.0435 × 10²⁴ kg

Part (a)

Step 1✍️ By handRound

\dfrac{5.97\times10^{24}}{7.35\times10^{22}} = 81.22\ldots

\approx 81.2

timesA1

Part (b)

Step 1✍️ By handCheck the sum by hand

597\times10^{22} + 7.35\times10^{22} = 604.35\times10^{22}

= 6.0435\times10^{24} \approx 6.04\times10^{24}

kgA1

Final answer (a)

81.2

times (b)

6.04\times10^{24}

Question 9●●● Exam-level✍️ + 🖩 · Paper 2[6 marks]

In one year a diamond mine near Orapa produces

2.45\times10^{7}

carats of diamonds. One carat is

0.2

grams. The diamonds are sold for a total of P

4.6\times10^{10}

(a) Find the total mass of the diamonds in kilograms. Give your answer in standard form.[2]
(b) Find the average price per carat, in pula, giving your answer in the form $a\times10^{k}$ , where $1\le a\lt 10$ , to 3 significant figures.[2]
(c) Botswana has a population of about $2.4\times10^{6}$ . Find the value of the diamonds per person, in standard form, to 3 significant figures.[2]

Show the full step-by-step solution

Step 1🧠 ThinkPlan each part

(a) carats × 0.2 gives grams; then divide by 1000 (

10^{3}

) for kilograms. (b) total price ÷ number of carats. (c) total price ÷ number of people.

Part (a)

Step 1✍️ By handGrams first, then kilograms

2.45\times10^{7}\times0.2 = 0.49\times10^{7} = 4.9\times10^{6}

gM1

4.9\times10^{6}\div10^{3} = 4.9\times10^{3}

kgA1

Step 2🖩 On the GDCParts (b) and (c) on the GDC

4.6›EE💻 click EE on keypad›10›÷💻 /›2.45›EE💻 click EE on keypad›7›enter💻 Enter›4.6›EE💻 click EE on keypad›10›÷💻 /›2.4›EE💻 click EE on keypad›6›enter💻 Enter

Line 1 checks part (a) in grams

Part (b)

Step 1✍️ By handPrice per carat

\dfrac{4.6\times10^{10}}{2.45\times10^{7}}

\approx \text{P }1.88\times10^{3}

per caratA1

Part (c)

Step 1✍️ By handValue per person

\dfrac{4.6\times10^{10}}{2.4\times10^{6}}

\approx \text{P }1.92\times10^{4}

per personA1

Final answer (a)

4.9\times10^{3}

kg (b) P

1.88\times10^{3}

1.92\times10^{4}

Question 10●●● Exam-level🖩 GDC · Paper 2[6 marks]

The star Proxima Centauri is about

4.0\times10^{13}

km from Earth. Light travels at

3.00\times10^{5}

km per second. Take one year to be 365.25 days.

(a) Find the time, in seconds, for light from Proxima Centauri to reach Earth. Give your answer in standard form.[2]
(b) Convert your answer to (a) into years, to 3 significant figures.[2]
(c) A space probe travels at $6.1\times10^{4}$ km per hour. Find how many years it would take to reach Proxima Centauri. Give your answer in standard form, to 3 significant figures.[2]

Show the full step-by-step solution

Step 1🧠 ThinkTime = distance ÷ speed

The units must match: km ÷ (km per second) gives seconds; km ÷ (km per hour) gives hours. A year has

365.25\times24\times3600

seconds and

365.25\times24

hours.

Step 2🖩 On the GDCUse ans so you never round too early

4›EE💻 click EE on keypad›13›÷💻 /›3›EE💻 click EE on keypad›5›enter💻 Enter›ans/(365.25·24·3600)›enter💻 Enter

Seconds, then years; hours, then years

Part (a)

Step 1✍️ By handSeconds

t = \dfrac{4.0\times10^{13}}{3.00\times10^{5}}

= 1.33\times10^{8}

sA1

Part (b)

Step 1✍️ By handYears

\dfrac{1.333\ldots\times10^{8}}{365.25\times24\times3600}

\approx 4.23

yearsA1

Part (c)

Step 1✍️ By handThe probe

\dfrac{4.0\times10^{13}}{6.1\times10^{4}} = 6.557\ldots\times10^{8}

hoursM1

\div(365.25\times24) \approx 7.48\times10^{4}

yearsA1

Final answer (a)

1.33\times10^{8}

s (b)

4.23

years (c)

7.48\times10^{4}

years

Question 11●●● Exam-level✍️ By hand · Paper 1 · no calculator[6 marks]

Let

x = 1.2\times10^{n+2}

and

y = 3\times10^{n}

, where

n\in\mathbb{Z}

(a) Show that $x + y = 1.23\times10^{n+2}$ .[2]
(b) Find $\dfrac{x}{y}$ , giving your answer in standard form.[2]
(c) Find $x^{2}$ , giving your answer in the form $a\times10^{k}$ , where $1\le a\lt 10$ and $k$ is an expression in $n$ .[2]

Show the full step-by-step solution

Step 1🧠 ThinkMake the powers match before adding

10^{n+2}

10^{2}\times10^{n} = 100\times10^{n}

. So

x = 1.2\times100\times10^{n} = 120\times10^{n}

. Now

x

and

y

both use

10^{n}

Part (a)

Step 1✍️ By handShow that (end with AG)

x + y = 120\times10^{n} + 3\times10^{n}

= 123\times10^{n}

= 1.23\times10^{2}\times10^{n} = 1.23\times10^{n+2}

Part (b)

Step 1✍️ By handDivide: subtract the powers

\dfrac{x}{y} = \dfrac{1.2}{3}\times10^{(n+2)-n} = 0.4\times10^{2}

= 4\times10^{1}

Part (c)

Step 1✍️ By handSquare: double the power

x^{2} = 1.2^{2}\times10^{2(n+2)}

= 1.44\times10^{2n+4}

💡 Remember In a “show that” question the answer is given. Marks come from the working, so every line must follow clearly from the one before.

Final answer (a) shown (b)

4\times10^{1}

(that is, 40) (c)

1.44\times10^{2n+4}

🆘 Stuck on a question? Open the notes, GDC toolkit and common mistakes

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

🎯 What you must know

Standard form: $a\times10^{k}$ where $1\le a\lt 10$ and $k$ is an integer. So $52\,300\,000 = 5.23\times10^{7}$ and $0.000\,407 = 4.07\times10^{-4}$ .
Big numbers have a positive power of 10; numbers between 0 and 1 have a negative power.
Multiply: multiply the front numbers, add the powers: $(a\times10^{m})(b\times10^{n}) = ab\times10^{m+n}$ .
Divide: divide the front numbers, subtract the powers: $\dfrac{a\times10^{m}}{b\times10^{n}} = \dfrac{a}{b}\times10^{m-n}$ .
Add or subtract: first rewrite both numbers with the same power of 10, then add the front numbers.
After any operation, tidy up so that the front number is between 1 and 10: $12\times10^{3} = 1.2\times10^{4}$ , $0.5\times10^{6} = 5\times10^{5}$ .
On the TI-Nspire, the EE key types “ $\times10^{\square}$ ”. The answer 2.99E−23 means $2.99\times10^{-23}$ — never write “E” in your final answer.

Standard form

a\times10^{k},\quad 1\le a\lt 10,\ k\in\mathbb{Z}

🧠 Learn this one

Multiplying

10^{m}\times10^{n} = 10^{m+n}

🧠 Learn this one

Dividing

\dfrac{10^{m}}{10^{n}} = 10^{m-n}

🧠 Learn this one

🖩 Your GDC toolkit for this subtopic

Follow the steps on your own TI-Nspire CX II. The screens show what you should see.

Type a number in standard form, e.g. 3.2 × 10⁵

3.2›EE💻 click EE on keypad›5

Type 3.2.
Press EE (it types E, which means “× 10 to the power”).
Type 5.

Type a negative power, e.g. 4.5 × 10⁻²

4.5›EE💻 click EE on keypad›(−)💻 type -›2

Type 4.5.
Press EE (it types E, which means “× 10 to the power”).
Press (−) (the negative key, not the minus key).
Type 2.

Make answers appear in standard form: doc → Settings → Document Settings → Exponential Format → Scientific

Press doc.
Choose Settings, then Document Settings.
Press tab until Exponential Format is highlighted, press click and choose Scientific.
Press tab to OK and press enter.

Set Exponential Format to Scientific

⚠️ Where students lose marks

Leaving the answer as $12\times10^{3}$ or $0.4\times10^{2}$ . This is not standard form — the front number must be at least 1 and less than 10.
Adding the powers when you add two numbers: $6\times10^{4} + 5\times10^{3}$ is not $11\times10^{7}$ .
Moving the decimal point the wrong way when tidying up: making the front number bigger makes the power smaller.
Copying “E” from the calculator screen into the answer. Write $\times10^{k}$ .
Forgetting brackets on the GDC: $4\times10^{13}\div(6.1\times10^{4})$ — with EE you are safe, but typed powers of 10 need brackets.

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL AA 1.1 — Operations with numbers in standard form

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL

1.1 Operations with numbers in standard form

✏️ Practice questions

🎯 What you must know

🖩 Your GDC toolkit for this subtopic

⚠️ Where students lose marks