Zero and negative exponents: extending the rule rather than inventing a meaning

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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 and exponent
The laws of exponents, and where each one comes from — the product and power-of-a-power rules, and the habit of expanding a power into its factors before reasoning about it
Cancelling equal factors from the top and bottom of a fraction
Reciprocals: that 1 over a number, multiplied by that number, gives 1
Subtraction that crosses zero, and reading a negative result

What they should be able to do

Derive the division rule for powers of a shared base by cancelling factors
State the conditions the chapter attaches to that rule, and say why a zero base is excluded
Explain why the value of 2⁰ is forced rather than chosen
State that any nonzero base raised to the exponent 0 gives 1, and justify it
Convert between a negative exponent and a reciprocal, in both directions
Evaluate expressions mixing positive and negative exponents of one base
Simplify a product or quotient of powers where the exponents are letters
Say what question the chapter leaves open at the end of this section, and why it matters

Where it usually goes wrong

"2⁰ is 0, because there is nothing to multiply." The empty product is 1, not 0, and the reason is the rule, not a convention about emptiness. Run 2⁴ ÷ 2⁴: nobody disputes that a number divided by itself is 1.
"2⁻¹ is −2." The minus sits on the exponent, not on the value. A negative exponent of a positive base never produces a negative number. Put 2⁻¹ = ½ and −2 side by side.
"A negative exponent means you divide by the base once." It means you divide as many times as the exponent says, starting from 1. 2⁻⁶ is one sixty-fourth, not one half.
"0⁰ is 1 like everything else." The chapter's rules exclude a base of 0 every time, and the reason is exactly here: the derivation runs through division by xᵃ, which is illegal when x is 0. Answer the chapter's own Math Talk question rather than skipping it.
"You can only divide a bigger power by a smaller one." That restriction is in the chapter's first statement of the rule, and the whole of this topic is about lifting it. Show why it was there and how it goes.
"These are new definitions to memorise alongside the old ones." They are the only values consistent with the rule already proved. That is the argument, and an explanation that states 2⁰ = 1 without it has taught a fact instead of a reason.
"1 over 10⁻³ needs a new rule." It needs the reciprocal of a reciprocal, which the chapter works out in two lines. Show the working.

Questions to check understanding

Simplify a quotient of two powers of one base, including cases where the answer has a zero or negative exponent
Write a given negative-exponent expression without a negative exponent, and the reverse
Evaluate a small negative power exactly, as a fraction
Simplify a product mixing positive and negative exponents of the same base
Simplify an expression whose exponents are letters
State the conditions under which each rule holds, and give a case where a condition fails
Explain why 2⁰ must be 1 — the justification-style question, not the value
Decide whether a stated manipulation is legal and say which rule licenses it

Examples worth working on the board

The halving picture (Part I p.27). Start with a line 16 units long, which the chapter writes as 2⁴. Erase half and 2⁴ ÷ 2 leaves three 2s, that is 2³ = 8 units. Erase half again: 2⁴ ÷ 2² leaves two 2s, so 2² = 4 units. A third halving: 2⁴ ÷ 2³ leaves one 2, so 2¹ = 2 units. Each step is shown as a fraction with the factors cancelling.
The pattern read off (Part I p.27): 2⁴ ÷ 2³ can be written 2⁴⁻³, giving 2¹.
An open prompt (Part I p.27): express 2¹⁰⁰ ÷ 2²⁵ as a power of 2.
The rule as printed (Part I p.27): nᵃ ÷ nᵇ = nᵃ⁻ᵇ, subject to n not being 0, with a and b both counting numbers and a larger than b.
A Math Talk challenge (Part I p.28), left unanswered: why is a base of 0 ruled out?
The zero exponent, forced (Part I p.28). The chapter observes it has not covered an exponent of 0, and asks what 2⁰ is. It then fixes the value so that the rule already established stays true: 2⁰ = 2⁴⁻⁴ = 2⁴ ÷ 2⁴, and writing four 2s over four 2s and cancelling gives 1. The same argument with a letter: 2⁰ = 2ᵃ⁻ᵃ = 2ᵃ ÷ 2ᵃ = 1.
The general statement (Part I p.28): xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰, so 1 = x⁰, provided x is not 0, with a any counting number.
The cartoon panel "When Zero is in Power!" (Part I p.28, artwork). Six wordless panels of a two-pan balance. Red numeral characters occupy one pan and blue numeral characters the other; across the panels the red group is jostled, crushed and thinned out, until in the last panel it is flung clear of the pan altogether and only the blue group is left, its pan swung right down. The individual digits are drawn as cartoon faces and I could not read them reliably even on the printed page, so this brief does not state them. The gag is decorative; nothing in the mathematics depends on it, and the explanation does not need to reproduce it.
Halving past the end (Part I p.28). Take the same 2⁴-unit line and halve it five times. Written as a fraction, four 2s over five 2s cancels to one half. Written by the rule, 2⁴ ÷ 2⁵ = 2⁽⁴⁻⁵⁾ = 2⁻¹. So 2⁻¹ is a half.
Ten halvings (Part I p.28). The same line halved ten times gives 2⁴ ÷ 2¹⁰ = 2⁽⁴⁻¹⁰⁾ = 2⁻⁶ units. Expanded, four 2s over ten 2s leaves 1 over 2⁶, which is one sixty-fourth. So 2⁻⁶ names 1/64.
Two more instances (Part I p.29): 10⁻³ is 1 over 10³, and 7⁻² is 1 over 7².
The reciprocal read backwards (Part I p.29). Can 10³ be written as 1 over 10⁻³? The chapter works it: 1 over 10⁻³ is 1 divided by 1/10³, which is 1 × 10³, which is 10³. Similarly 7² is 1 over 7⁻², and 4ᵃ is 1 over 4⁻ᵃ.
The two-part general form (Part I p.29): n⁻ᵃ = 1 over nᵃ, and nᵃ = 1 over n⁻ᵃ, in both cases with n not equal to 0.
The three rules boxed together (Part I p.29), in one row: the product rule, the power-of-a-power rule, and the division rule.
The open question (Part I p.29, flagged Math Talk). The rules were stated for counting numbers. Can the two exponents be any integers instead, and would the rules survive? The chapter poses this and does not answer it here.
Exercise set A (Part I p.29): write equivalent forms of 2⁻⁴; 10⁻⁵; (−7)⁻²; (−5)⁻³; and 10⁻¹⁰⁰.
Exercise set B (Part I p.29), to be simplified into exponential form: 2⁻⁴ × 2⁷; 3² × 3⁻⁵ × 3⁶; p³ × p⁻¹⁰; 2⁴ × (−4)⁻²; and 8ᵖ × 8ᑫ.

Figures to have open

A halving bar — one strip, repeatedly halved, with the exponent counter falling 4, 3, 2, 1, 0, −1, −2 alongside. This single figure carries sections 1 to 9 and is the only one the topic genuinely needs. The chapter has the idea in words at Part I pp.27–28 but draws no such strip; this is added here.
A cancellation frame showing equal factors struck out above and below a bar. Standard schematic, but it must be legible when the counts are equal, so the frame has to survive "everything cancels".
A two-column identity card for n⁻ᵃ and 1/nᵃ, so the reciprocal relationship reads in both directions. Standard schematic.
The chapter's balance-scale cartoon (Part I p.28) is not required and should not be reproduced.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 2, "Power Play", §2.3 "The Other Side of Powers", Part I pp.27–29, up to the unnumbered subheading "Power Lines". This span includes the unnumbered subheading "When Zero is in Power!" at Part I p.28.
Two Math Talk flags fall inside this span: the question about a zero base at Part I p.28, and the question about integer exponents at Part I p.29.
The rules being extended were derived at Part I pp.24–25 (The laws of exponents, and where each one comes from). The line of powers that displays the extension is the next topic, Reading a power line: multiplication as movement along a scale.
The chapter's SUMMARY (Part I p.46) lists n⁻ᵃ as 1 over nᵃ and n⁰ as 1 for nonzero n, alongside the other four rules.
Chapter-end exercises reaching this material: "Figure it Out" at Part I p.44 items 3 and 5, and Part I p.45 item 7.