Exponential notation: repeated multiplication written once

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What to assume they know

Paper folding: the growth that outruns intuition — the folded sheet, and doubling as repeated multiplication
What makes a number a perfect square and What makes a number a perfect cube, and the three-identical-groups test — square and cube numbers, and the notations n² and n³
Prime factorisation of a number up to five digits by repeated division
Multiplying negative integers, and the sign of a product of several negatives
Using a letter to stand for an unknown number in an expression

What they should be able to do

Rewrite a product of equal factors in exponential form, and expand an exponential form back into a product
Name the base and the exponent of a given power, and read a power aloud in the ways the chapter accepts
Write a number's prime factorisation in exponential form after a division ladder
Evaluate a power with a negative base and predict the sign from the exponent
Distinguish repeated addition from repeated multiplication for the same pair of numbers
Write a product of powers of two different letters, such as a³b², and read it aloud
Count the leaves of a branching structure in which each node splits the same number of ways, and express the count as a power
Count the settings of a lock with a fixed number of slots and options, and express the count as a power
Say what a power counts in a given situation, rather than only computing it

Where it usually goes wrong

"The exponent tells you how many times to multiply." It tells you how many factors there are. Multiplying is done one time fewer than that, which is why 2¹ is 2 and not 4. "How many 2s" is the phrasing that keeps a student out of this trap, so prefer it when explaining. But do not tell them the other phrasing is wrong: the chapter itself writes nᵃ as n multiplied by itself a times, at Part I p.22 and again in the SUMMARY at Part I p.46, and that is the wording an exam will use. Teach the safer form, and say plainly that the book's form means the same thing once you count factors rather than operations.
**"nᵃ means n × a."** The chapter puts the trap on the page as options (i) and (iii) of the six-option question, and again in the line contrasting 4 + 4 + 4 with 4 × 4 × 4. Show both computed.
"A negative base makes a negative answer." Only for an odd exponent. The chapter asks for (−1)⁵ and (−1)⁵⁶ side by side precisely so the parity of the exponent, not the sign of the base, is what decides.
"(−2)⁴ and −2⁴ are the same thing." They are not, and the chapter's request to verify (−2)⁴ = 16 is the place to separate them. Brackets are load-bearing.
"Prime factorisation in exponential form is a different factorisation." It is the same list of primes, counted. 32400 has ten prime factors either way.
"The tree's answer is 3 × 4." Four levels of three-way branching give 3⁴, not 12. The tree is the figure that makes this impossible to get wrong, which is why it must be drawn and not described.
"A lock with more slots is safer than a lock with more symbols." Both help, but not equally — that is exactly the comparison Estu's 6-slot letter lock invites against the 5-slot digit lock. Leave it as a question, not a verdict.
"1,00,000 passwords means 1,00,000 tries." The chapter says the lock opened on the last one, so the count of tries equals the count of passwords only in the worst case. Worth one sentence.

Questions to check understanding

Express a given product of equal factors in exponential form, including products mixing two or three letters
Give the prime factorisation of a 3- or 4-digit number in exponential form
Evaluate an expression combining two powers, at least one with a negative base
Identify base and exponent in a given power and read it aloud correctly
Choose the correct expression for a described repeated-doubling situation from a list of near-misses
Count the outcomes of a repeated choice — passwords, number plates, seating — and express the count as a power
Count the leaves of a uniform branching structure described in words
Explain the difference between na and nᵃ using a worked pair

Examples worth working on the board

The folded sheet, rewritten (Part I p.21). One fold gives 0.001 cm times 2, which is 0.002 cm. Two folds give 0.001 cm times 2 times 2, written 0.001 cm × 2², coming to 0.004 cm. Three folds give 0.001 cm × 2³ = 0.008 cm. Four folds give 0.001 cm × 2⁴ = 0.016 cm. Seven folds give 0.001 cm × 2⁷ = 0.128 cm. All printed.
The reading ladder (Part I pp.21–22). n × n is n²; three factors give n³; four give n⁴; seven give n⁷; and in general a factors give nᵃ.
The worked power (Part I p.22): 5⁴ = 5 × 5 × 5 × 5 = 625. Here 4 is the exponent, 5 is the base, and 625 has 5⁴ as its exponential form. The chapter sets a tinted box beside it giving four accepted readings of 5⁴ — raised to the power 4, to the power 4, power 4, and 4th power of 5.
Powers of 5 as a family (Part I p.22): expressions 5ⁿ are called powers of 5, and 5¹, 5², 5³, 5⁴ are listed as instances.
1024 comes back (Part I p.22): ten 2s multiplied together give 2¹⁰ = 1024, which is the same 1024 that ten folds produced in §2.1.
The six-option question (Part I p.22). With the starting thickness written as the letter-number v, which expression is the thickness after ten folds? The options printed are 10v; 10 + v; 2 × 10 × v; 2¹⁰; 2¹⁰v; and 10²v.
Two small examples (Part I p.22): 4 × 4 × 4 = 4³ = 64, and (−4) × (−4) × (−4) = (−4)³ = −64.
Two letter examples (Part I p.22): the product of three as with two bs is written a³b², which the chapter reads aloud as a cubed, then b squared; the product of two as with four bs is written a²b⁴.
The warning line (Part I p.22): three 4s added come to 12, whereas three 4s multiplied come to 64.
The division ladder for 32400 (Part I p.22, set down the right margin). Divide by 2 four times — 32400, 16200, 8100, 4050 — then by 5 twice — 2025, 405 — then by 3 four times — 81, 27, 9, 3 — ending at 1. The chapter prints 32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3, and in exponential form 32400 = 2⁴ × 5² × 3⁴.
Four open prompts (Part I p.22), all left unanswered by the chapter: is (−1)⁵ positive or negative, and what about (−1)⁵⁶; is (−2)⁴ equal to 16, to be verified; what are 0² and 0⁵; and what is 0ⁿ.
Figure it Out, item 1 (Part I p.22). Six products to put in exponential form: four 6s; two ys; four bs; two 5s together with three 7s; two 2s together with two as; and three as together with four cs and a single d.
Figure it Out, item 2 (Part I p.23). Write each number as primes raised to powers: 648; 405; 540; 3600.
Figure it Out, item 3 (Part I p.23). Find the value of: 2 × 10³; 7² × 2³; 3 × 4⁴; (−3)² × (−5)²; 3² × 10⁴; and (−2)⁵ × (−10)⁶. Items four and six are the sign question in disguise.
The verse and the tree, "The Stones that Shine ..." (Part I p.23). The eight-line verse sets up a chain of threes: a king, three daughters, three baskets each, three keys in each basket, three rooms opened by each key, three tables in each room, three necklaces on each table, three diamonds on each necklace. A hint tells the student to count baskets and rooms first.
The printed tree diagram (Part I p.23, artwork). Drawn as a single node at the top labelled King, fanning into three panels — one per daughter — and four labelled levels down the right-hand edge: daughters, baskets, keys, rooms. The drawing stops at the room level; the last three levels of the verse are not drawn. Verified on the printed page.
Counting up the tree (Part I p.23): the room count is 3⁴, reached by repeated multiplication — 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81, then 81 × 3 = 243. Note that the printed chain runs one step past the room level. The diamond count is then stated as seven 3s multiplied, that is 3⁷.
The combination warm-up (Part I p.26). Estu has 4 dresses and 3 caps. Per cap there are 4 dresses, giving 4 + 4 + 4 = 4 × 3 = 12; read the other way, per dress there are 3 caps, giving 3 + 3 + 3 + 3 = 3 × 4 = 12. An illustration above is drawn as two panels, one per reading: the left fans four arrows down from a single cap to four garments, and the right fans three arrows up from a single garment to three caps. Only one of the two fans points downward.
The wardrobe question (Part I p.26): Roxie owns 7 dresses, 2 hats, 3 pairs of shoes. In how many ways can she dress? A hint suggests drawing the same kind of diagram. Left open.
The safe (Part I p.26). Estu and Roxie find a safe of old stamps and coins left by their great-grandfather, locked by a 5-digit password. They try every password and the lock opens on the last one. How many did they check?
The simpler-version method (Part I p.26), set in a tinted panel with an owl: when a problem resists, solve a smaller version of it first.
Building up to five slots (Part I pp.26–27). A 2-digit lock: 10 choices for the first digit, 10 for the second, so 10 × 10 = 100. A 3-digit lock: every one of those 100 two-digit passwords admits 10 third digits, so 100 × 10 = 1000. A 5-digit lock: 10 × 10 × 10 × 10 × 10 = 10⁵ = 1,00,000, which the chapter notes is the same as writing every 5-digit string from 00000 to 99999.
The dial figure (Part I p.27, artwork). Two vertical columns of ten boxes each, both running 0 to 9 — left column blue, right column green. Ten lines fan out from the single left-hand box labelled 0, one to each right-hand box, so the drawing shows the ten choices open after one digit is fixed, not the full hundred. Verified on the printed page.
Estu's proposed lock (Part I p.27): 6 slots, each carrying the letters A to Z, which he thinks safer. How many passwords does that allow? Left open; the chapter does not state the number of letters.
Try This (Part I p.27): think about combination counts in pincodes, mobile numbers and vehicle registration numbers, and find out how they are allotted. Two pincodes are printed as examples — Vidisha in Madhya Pradesh, 464001, and Zemabawk in Mizoram, 796017.

Figures to have open

The king's tree, redrawn as a clean branching diagram with the level names down one side and the running count 3, 3², 3³, 3⁴ down the other. The chapter's own figure (Part I p.23) stops at rooms. Central to sections 10 and 11.
The two-column dial figure for a 2-digit lock (Part I p.27), then the same idea extended a slot at a time. Standard schematic.
A factor strip showing 32400's ten primes sorted into groups, with the group sizes lifted into exponents. An added figure.
A caps-and-garments grid for Estu's 12 combinations, drawn as a rectangle so that the two readings — per cap and per dress — are the rows and columns of one array. The chapter draws arrows rather than a grid; the grid is the better figure and is added here.
No photograph is needed.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 2, "Power Play", §2.2 "Exponential Notation and Operations", Part I pp.21–23 for the notation itself, and the unnumbered subheadings "The Stones that Shine ..." (Part I p.23) and "How Many Combinations" (Part I pp.26–27).
Exercise items: "Figure it Out" at Part I p.22 item 1 and Part I p.23 items 2 and 3.
The rules that come out of the same subsections are the business of The laws of exponents, and where each one comes from; this topic stops at the notation and at what a power counts.
The 1024 that reappears at Part I p.22 was established at Part I p.21 (Paper folding: the growth that outruns intuition).
The chapter's SUMMARY (Part I p.46) restates nᵃ as n multiplied by itself a times.