The laws of exponents, and where each one comes from

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

Exponential notation: repeated multiplication written once — exponential form, base, exponent, and what a power counts
That multiplication can be regrouped and reordered without changing a product
Multiplication tables through 81 × 27, and confident work with 64 × 64
Working with a letter as a base (p⁴) and as an exponent (nᵃ)
The idea of a counting number, since every law here is stated only for those

What they should be able to do

Split a power into a product of two powers of the same base, in more than one way, and justify the split
State and apply the rule for multiplying powers that share a base
Rewrite a single power so that its base is itself a power, in two or more distinct ways
Show that the two exponents of a power of a power may be exchanged, and say why
State and apply the rule for multiplying powers that share an exponent
State the matching rule for dividing powers that share an exponent
Re-express a product of unequal powers, such as 2⁴ × 3⁴, as a single power
Identify which of these rules a given expression needs, and in what order
Say precisely which numbers each rule has so far been established for

Where it usually goes wrong

**"nᵃ × nᵇ = nᵃᵇ."** The single most common error in the chapter's material, and it is settled by expanding: four 3s next to three 3s is seven 3s, not twelve. Expand before you assert.
"2⁴ × 3⁴ = 6⁸." The bases combine; the exponent does not. Damayanti's ponds exist to make this concrete — the flowers were counted once, so there are four pairings, not eight.
"2³ × 3⁴ can be combined too." It cannot, by either law: the bases differ and so do the exponents. Students who learn the third law without its condition apply it everywhere. Show a case where nothing can be merged.
"(4³)² means 4³ times 2." It means 4³ multiplied by itself. Write the six 4s out once and the confusion does not survive.
"(4³)² and (4²)³ are different numbers." They are the same six factors, bracketed differently. Both come to 4096 on the printed page.
"The pond is half covered on day 15." Half of thirty days is not half the pond. The count doubles daily, so the pond fills its second half in the final day. This is the chapter's own trap, and it works on adults.
"The laws are definitions to memorise." Every one of them was derived on the page from a picture of the factors. An explanation that lists the three rules and moves on has taught the by-product and skipped the content.
"These laws hold for any exponents at all." Not yet. The chapter is careful to say counting numbers, and only asks the wider question at Part I p.29.

Questions to check understanding

Simplify a product of two or three powers sharing a base, giving the answer in exponential form
Rewrite a given power so that its base is itself a power, in two ways
Decide whether two given expressions are equal, and justify by expanding
Combine a product of two powers with a shared exponent into a single power
Simplify a quotient of two powers with a shared exponent
Identify the error in a worked line that has added bases or multiplied exponents
Apply the rules to expressions in letters, including a mixed product such as m⁵n¹²(mn)⁹
Explain in words why the product rule holds — the reasoning-style question the board now favours over the manipulation

Examples worth working on the board

Splitting the diamond count (Part I p.24). The chapter writes 3⁷ as a bracketed four followed by a bracketed three, that is 3⁴ × 3³. Since the tree work had already reached 3⁴ = 81 and 3³ = 27, the product is 81 × 27 = 2187. The page also draws a run of seven 3s with two braces under it, one labelled 3⁴ and the other 3³.
The second split, left open (Part I p.24): 3⁷ can equally be written as 3² × 3⁵, and the student is asked to reason out why. Do not hand over the reason; build it from the row of seven.
The letter case (Part I p.24): p⁴ × p⁶ expands into four ps followed by six ps, so it is p¹⁰.
The first law as printed (Part I p.24): nᵃ × nᵇ = nᵃ⁺ᵇ, with a and b both counting numbers.
A Math Talk prompt (Part I p.24): use that observation to compute 2⁹, 5⁷ and 4⁶. Left open.
4⁶ two ways (Part I p.24), set as two side-by-side boxes.
- Left box: three 4s times three 4s is 4³ × 4³ = 64 × 64 = 4096. Since 4³ × 4³ is the square of 4³, it may also be written (4³)².
- Right box: two 4s, three times over, is 4² × 4² × 4² = 16 × 16 × 16 = 4096. Since that is the cube of 4², it may also be written (4²)³.
Two more instances (Part I p.24): 7⁴ regroups into two pairs of 7s, giving 7² × 7², that is (7²)². And 2¹⁰ regroups into five pairs, giving five copies of 2², that is (2²)⁵.
The swap, worked (Part I p.24). Asked whether 2¹⁰ also equals (2⁵)², the chapter regroups the ten 2s into two blocks of five, gets 2⁵ × 2⁵, and so (2⁵)². The same ten factors, blocked two different ways.
The second law as printed (Part I p.24): (nᵃ)ᵇ = (nᵇ)ᵃ = nᵃ˟ᵇ = nᵃᵇ, with a and b both counting numbers.
A prompt on the second law (Part I p.24): rewrite 8⁶, 7¹⁵, 9¹⁴ and 5⁸ so that each has a power for its base, giving two or more versions of every one. Left open.
The Magical Pond, part one (Part I p.25). A pond holds one pink lotus; the lotus count doubles daily; on day 30 the pond is fully covered. On which day is it half covered? The chapter answers this one — day 29 — and then asks for the lotus count in exponential form for both the full pond and the half pond, which it leaves open.
The Magical Pond, part two (Part I p.25). A second pond triples daily. Both ponds start empty; Damayanti drops a single lotus into the doubling one. Four days later she carries every flower across to the tripling pond and waits 4 more days. The chapter computes: the first 4 days give 1 × 2 × 2 × 2 × 2 = 2⁴; the next 4 days multiply that by four 3s, giving 2⁴ × 3⁴.
Reversing the order (Part I p.25). Had she used the tripling pond first, the count would be 1 × 3⁴ × 2⁴, which the chapter writes out as a run of four 3s followed by a run of four 2s. Regrouped into four brackets of (3 × 2), it becomes (3 × 2)⁴ = 6⁴.
The third law as printed (Part I p.25): mᵃ × nᵃ = (mn)ᵃ, with a a counting number. Two prompts follow, both left open: compute 2⁵ × 5⁵, and simplify 10⁴ over 5⁴ into exponential form.
The division twin (Part I p.25): the chapter closes the subsection by stating that mᵃ over nᵃ can be shown to equal (m over n)ᵃ. It gives no derivation here; the SUMMARY at Part I p.46 repeats it with the condition that the lower base is not zero.
The pond illustration (Part I p.25, artwork). A seated figure beside a circular pond covered in lotus pads and pink blooms. Decorative; the counting is not readable from it.

Figures to have open

The factor-row figure — a horizontal row of identical factors with movable braces beneath it. One figure carries sections 1 to 8: slide the braces and every law in this topic appears. The chapter draws a static version of it at Part I p.24; the explanation's version has to be movable, because the argument is that the brace position is free.
The zip diagram for the third law: a row of 3s above a row of 2s, vertical links pairing them, resolving into a row of 6s. An added figure; the chapter does the regrouping in symbols only.
A two-pond timeline with a day axis, one lane doubling and one tripling, and the transfer marked at day 4. Standard schematic.
No photograph is needed; the chapter's lotus artwork is decorative.

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.23–25 — the closing part of the unnumbered subheading "The Stones that Shine ..." and the whole of "Magical Pond".
The three laws are restated together in a boxed row at Part I p.29, and again in the chapter's SUMMARY at Part I p.46.
Exercise items using these rules appear in the chapter-end "Figure it Out" at Part I pp.44–45, in particular items 3, 4, 5 and 7.
The counting structures these products came from — the king's tree and the lock — belong to Exponential notation: repeated multiplication written once. The division rule, and what happens once the exponents are allowed to be zero or negative, belong to Zero and negative exponents: extending the rule rather than inventing a meaning.