Power Play
Class 8 Mathematics · Ganita Prakash Part 1 · Chapter 2
2- Mathematics
- Mathematics
- Science
- + 3 more
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
- Space weighted by chapter size
- Charts from the book's own data
- Traps corrected
Summary
Power Play extends squares and cubes to general exponents and shows the astonishing pace of multiplicative (exponential) growth — for instance, a paper folded \(46\) times would reach the Moon. The notation \(n^a\) means \(n\) multiplied by itself \(a\) times; \(n\) is the base and \(a\) the exponent. The laws of exponents are developed from first principles: \(n^a \times n^b = n^{a+b}\), \((n^a)^b = n^{ab}\), \(n^a \div n^b = n^{a-b}\), \(m^a \times n^a = (mn)^a\), and \(\dfrac{m^a}{n^a} = \left(\dfrac{m}{n}\right)^a\). To keep the division law consistent, \(n^0 = 1\) for \(n \ne 0\), and negative exponents emerge as \(n^{-a} = \dfrac{1}{n^a}\). Powers of ten let us write numbers in expanded form and, more powerfully, in scientific (standard) notation \(x \times 10^y\) with \(1 \le x < 10\), where the exponent matters more than the coefficient. The chapter uses this to compare gigantic real-world quantities — populations, distances, ages of the universe — and contrasts linear (additive) growth with exponential (multiplicative) growth.