S1 Maths | Index Notation and Scientific Notation: Index Laws, Powers of Ten Tutor Prep Guide

2026-11-14

Tutor lesson-prep guides · Secondary Maths

A student writes 2 cubed equals 6 because the raised 3 looks like a multiplier. Left alone, every index law after it collapses. This lesson expands the notation back into multiplication, then carries it to powers of ten and scientific notation.

Objectives — By the end students can: (1) state that 2 to the fifth is 32 and expand it as five twos multiplied, explaining why 2 cubed is not 6; (2) use the index laws to get 2 cubed times 2 to the fourth as 2 to the seventh, 128, and 5 to the sixth divided by 5 squared as 5 to the fourth, 625, and 3 squared cubed as 3 to the sixth, 729; (3) write 45000 as 4.5 times ten to the fourth and 0.00032 as 3.2 times ten to the minus fourth, explaining that the exponent counts decimal places moved.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Multiplication	What is 2 times 2 times 2?	8
Powers of ten	What is ten squared?	100
Decimal movement	Moving the decimal point one place right makes the number bigger or smaller?	Ten times bigger

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Start from the error	Write 2 cubed equals 6 and ask the class whether it is right, collecting opinions	Say what they think and why
5-14 min	An index is a count of factors	Expand 2 cubed as 2 times 2 times 2, which is 8, then 2 to the fifth is 32 and 3 to the fourth is 81	Expand 2 to the fourth and 5 to the zero aloud
14-24 min	Index laws	Expand 2 cubed times 2 to the fourth into seven twos to get 2 to the seventh, then 5 to the sixth over 5 squared and 3 squared cubed	Repeat: same base multiplying means add the indices
24-33 min	Scientific notation	Moving the point four places left in 45000 gives 4.5, so it is 4.5 times ten to the fourth; 0.00032 moves four right	Write 48000 and 0.0000705 themselves
33-41 min	Three independent items	Hand out an expansion, an index law and a scientific notation item, watching for 2 cubed written as 6	Expand once before answering
41-45 min	Exit check	Three oral questions: the value of 2 cubed, 2 cubed times 2 to the fourth, and 45000 in scientific notation	Answer aloud with a reason

03 Teaching script (read it out)

Scene 1: Act 1: the index counts factors, it does not multiply

“2 cubed means three twos multiplied: 2 times 2 times 2, which is 8. The 3 on top counts how many twos there are, it is not a multiplier, so 2 cubed is not 6. Try 2 to the fifth: five twos, so 2, 4, 8, 16, 32, and we write 2 to the fifth equals 32. For 5 to the zero, any non-zero number to the zero is 1, and the index laws force that answer to stay consistent.”

Students repeat: the raised number counts how many factors

Scene 2: Act 2: same base multiplies, so add the indices

“2 cubed times 2 to the fourth is three twos followed by four twos, seven twos in all, so 2 to the seventh, which is 128. Division runs the other way: 5 to the sixth over 5 squared leaves four fives, so 5 to the fourth, 625. The third law confuses most: 3 squared cubed is three copies of 3 squared multiplied, so the indices add to 2 plus 2 plus 2, giving 3 to the sixth, 729. All three laws are just counting how many factors you have.”

Students repeat: multiply add, divide subtract, brackets multiply

Scene 3: Act 3: scientific notation is counting decimal places

“To write 45000 as 4.5 times ten to the n, move the point from the end to just after the 4, four places to the left, so the exponent is 4 and it is 4.5 times ten to the fourth. Small numbers go the other way: 0.00032 moves four places right to sit after the 3, giving 3.2 times ten to the minus fourth. The rule is that the front number sits between 1 and 10, the exponent counts the places moved, positive for left and negative for right.”

Students repeat: between 1 and 10, left positive, right negative

04 Question types and worked steps

Type	Example	Steps	Answer
Expand an index	What is 2 cubed?	(1) Read the base and the index (2) write the chain of factors (3) multiply step by step (4) check the number of factors	2 times 2 times 2 is 8
Multiply and divide like bases	2 cubed times 2 to the fourth, and 5 to the sixth over 5 squared	(1) Confirm the bases match (2) add the indices to multiply (3) subtract to divide (4) state the result	2 to the seventh is 128; 5 to the fourth is 625
A power of a power	3 squared, all cubed	(1) The bracket holds one number (2) it is multiplied three times (3) multiply the indices (4) evaluate	3 to the sixth is 729
Scientific notation	Write 45000 as a times ten to the n	(1) Make the front number sit between 1 and 10 (2) count the places moved (3) positive for left, negative for right (4) write the final form	4.5 times ten to the fourth

Sanity check (teach this): Checks that must hold: an index result is larger than the base except for 0 and 1, so 2 to the fifth being 32 is sensible while 10 means a factor was missed; multiplying like bases must increase the value, so 128 is far above 2 to the fourth, 16; dividing must decrease it, so 625 is below 5 to the sixth, 15625; the front number in scientific notation must sit between 1 and 10, so 45 times ten cubed is not allowed; a positive power of ten exceeds 1, so 0.00032 written as 3.2 times ten to the fourth has the sign wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
Expand 2 to the fifth and give its value.	2 times 2 times 2 times 2 times 2 is 32	An answer of 10 means the index was treated as a multiplier
Work out 5 to the sixth divided by 5 squared and name the law used.	5 to the fourth, which is 625, subtracting indices for division	5 cubed means the subtraction was off by one
Write 0.0000705 in scientific notation.	7.05 times ten to the minus fifth, four plus one places right	A positive exponent means the direction was reversed

06 Common mistakes and what to say

Mistake	Student writes	Say this
Treating the index as a multiplier	“2 cubed is 2 times 3, which is 6.”	“The index counts the twos: 2 times 2 times 2 is 8. Count them, there are three, and 2 times 3 is a different sum.”
Multiplying the bases too	“2 cubed times 2 to the fourth is 4 to the seventh.”	“The base stays put and the indices add: three twos plus four twos is seven twos, so 2 to the seventh, 128.”
Adding the indices in a power of a power	“3 squared cubed is 3 to the fifth.”	“The bracket is multiplied three times, so 2 plus 2 plus 2 is 6, meaning the indices multiply and the answer is 3 to the sixth, 729.”
Front number outside 1 to 10	“45000 is 45 times ten cubed.”	“The front number must sit between 1 and 10, so it is 4.5 times ten to the fourth, and the point moves four places.”

07 Homework (paper, section and question numbers)

Redo Act 1: expand 2 cubed and 2 to the fifth, and explain why 2 cubed is not 6 — Chapter practice, question 1
Use the index laws on 2 cubed times 2 to the fourth, 5 to the sixth over 5 squared and 3 squared cubed, showing each step — Chapter practice, question 2
Preview: write 48000 and 0.0000705 in scientific notation and state which way the point moved — Chapter practice, question 3

08 Exit check (last 5 minutes)

Question	Answer	Decision
What is 2 cubed, and why?	8, because it is three twos multiplied	An answer of 6 means the index was a multiplier
What is 2 cubed times 2 to the fourth?	2 to the seventh, 128, adding indices for like bases	4 to the seventh means the bases were multiplied
What is 45000 in scientific notation?	4.5 times ten to the fourth, the point moving four left	45 times ten cubed breaks the 1 to 10 rule

09 Tutor FAQ

Why expand instead of moving straight to the index laws?

Once a student believes 2 cubed is 6, every later law is remembered for the wrong reason. One expansion takes two minutes and turns three laws into something derivable.

Why is 5 to the zero equal to 1?

Use division: 5 squared over 5 squared is 5 to the zero by the index law, and any number over itself is 1, so the zero power is 1. That derivation sticks better than a rule.

What is scientific notation actually for in an exam?

It lets very large and very small numbers be compared and multiplied, so 6 times ten to the seventh against 4 times ten to the fourth is visibly three orders of magnitude, which underpins estimation and unit conversion.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.