2. Power Play

Part I, printed Chapter 211 topics2 h

Chapter 2 of NCERT Mathematics for Class 8 (Part I, printed Chapter 2): Power Play. Four sections — Why powers grow so fast, and how to write them, The other side of powers, Powers of ten and Making sense of very large numbers. 11 videos, 2 h in all.

Worked answers to the 17 exercise questions in this chapter →

0 of 11 done — sign in to track this chapter

Part 1

Why powers grow so fast, and how to write them

10 minPaper folding: the growth that outruns intuitionFold a sheet of paper in half fifty times, with the physical limit imagined away, and it reaches the Sun. Intuition is not close.10 min · 12 partsहिंदी10 minExponential notation: repeated multiplication written onceWriting 2⁷ instead of seven 2s is not shorthand for the lazy. It promotes a number that was already in the problem into something you can compute with.10 min · 12 partsहिंदी9 minThe laws of exponents, and where each one comes fromNot one of the exponent laws is a new fact. Each says where you are allowed to put a bracket in a row of identical factors.9 min · 12 partsहिंदी

Part 2

The other side of powers

10 minZero and negative exponents: extending the rule rather than inventing a meaningNobody decided that 2⁰ should be 1. Keep erasing half of a line 16 units long and the value is forced on you.10 min · 12 partsहिंदी10 minReading a power line: multiplication as movement along a scaleStand all the powers of one number in an evenly spaced column and multiplication stops being an operation. It becomes a move.10 min · 12 partsहिंदी

Part 3

Powers of ten

10 minPowers of 10, and place value written out for whole numbers and decimalsExpanded form was a statement about powers of ten all along. Take 47561 apart the way you already do and the exponents are already there.10 min · 13 partsहिंदी11 minScientific notation, and why the standard form is 1 ≤ x < 10Keeping the front number between 1 and 10 is not tidiness. It is what makes the exponent worth trusting when you compare two quantities.11 min · 13 partsहिंदी

Part 4

Making sense of very large numbers

11 minEstimating a quantity nobody can count: guess, model, assume, approximateAn estimate is not a calculation done badly. It is a different object with a structure, and it starts from a question with no number in it at all.11 min · 13 partsहिंदी10 minAdditive growth versus multiplicative growthA ladder and a folded sheet both reach the Moon. One of them needs about forty million times as many steps.10 min · 13 partsहिंदी10 minWhy the nearest power of ten is the only handle on a quantity too big to picturePast a few thousand, the digits of a quantity carry no intuition. Its exponent does, because an exponent is a position on a ladder.10 min · 13 partsहिंदी10 minNaming the powers of ten: the Lalitavistara list, the million-to-decillion names, and the googolAncient India named the powers of ten far past anything anyone could count. The googol was named by a nine-year-old.10 min · 13 partsहिंदी

What you will be able to do

  • Continue a doubling table from a given starting value and fill missing entries
  • Rewrite a product of equal factors in exponential form, and expand an exponential form back into a product
  • Split a power into a product of two powers of the same base, in more than one way, and justify the split
  • Derive the division rule for powers of a shared base by cancelling factors
  • Build a line of the powers of a chosen base, with the exponents in order
  • Write a whole number in expanded form using the multipliers 10, 100, 1000
  • Convert a large whole number into scientific notation, and back
  • Write a relation among named quantities before attempting any arithmetic
  • Identify what is held constant in a described process — a fixed increase or a fixed multiplier
  • Express a stated quantity in scientific notation and name the power of ten nearest to it
  • Read a naming ladder and state the step it uses, in exponent terms

What this chapter assumes you already know

  • Multiplying and dividing decimals by 2, and by powers of 2
  • Reading a quantity across changing units — centimetres, metres, kilometres — and converting between them
  • Reading a two-column table where one column indexes the other
  • 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
  • That multiplication can be regrouped and reordered without changing a product
  • Multiplication tables through 81 × 27, and confident work with 64 × 64

It builds directly on What makes a number a perfect square and What makes a number a perfect cube, and the three-identical-groups test.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Each fold adds the same amount."
  • "Somebody dropped some zeros."
  • "You could really fold paper 46 times."
  • "The paper reaches the Moon, so it must be very long."
  • "≈ means the same as =."
  • "1024 is a coincidence."
  • "Growth this fast means the early folds were already big."
  • "The exponent tells you how many times to multiply."