Power Play

Chapter at a glance

• Thickness after repeated folding doubles each time, showing multiplicative (exponential) growth: after 46 folds the thickness exceeds 7,00,000 km.
• Any number can be written as \(n^a\), where \(n\) is the base and \(a\) is the exponent (power), meaning \(n\) multiplied by itself \(a\) times.
• Laws of exponents allow simplification of products, quotients and powers of powers.
• Negative exponents and zero exponent are defined so the laws remain consistent (\(n^0=1\), \(n^{-a}=\frac{1}{n^a}\), \(n\neq0\)).
• Numbers are written in scientific (standard) form as \(x\times10^y\) where \(1\leq x<10\).
• Exponential growth is far faster than linear (additive) growth; 46 folds reach the Moon while linear steps require billions.
• Powers of 10 express and compare extremely large quantities (populations, distances, time spans) concisely.
• Combinations and passwords are counted using products of choices, often expressed as powers (e.g., \(10^5\) five-digit codes).

Definitions, theorems and results

• \(n^a\) denotes the product of \(n\) multiplied by itself \(a\) times (\(a\) counting number). Read as “\(n\) raised to the power \(a\)”.
• \(n^0=1\) (\(n\neq0\)).
• Negative exponent: \(n^{-a}=\frac{1}{n^a}\) and \(n^a=\frac{1}{n^{-a}}\) (\(n\neq0\)).
• Product rule: \(n^a\times n^b=n^{a+b}\) (\(a,b\) counting numbers).
• Power-of-power rule: \((n^a)^b=(n^b)^a=n^{ab}\) (\(a,b\) counting numbers).
• Quotient rule: \(n^a\div n^b=n^{a-b}\) (\(n\neq0\), \(a>b\), \(a,b\) counting numbers).
• Product of like powers with different bases: \(m^a\times n^a=(m\times n)^a\) (\(a\) counting number).
• Quotient of like powers: \(m^a\div n^a=\left(\frac{m}{n}\right)^a\) (\(n\neq0\), \(a\) counting number).
• Scientific/standard form: any number written as \(x\times10^y\) where coefficient \(x\) satisfies \(1\leq x<10\) and exponent \(y\) is any integer.
• No formal theorems or lemmas requiring proof appear in the chapter; all results are introduced through patterns and examples.

Formula sheet

Solved-example patterns

• Express a product in exponential form: count identical factors for each base and write as base^exponent (e.g., \(a\times a\times b\times b\times b=a^2b^3\)).
• Prime-factorisation into exponential form: factorise completely, group identical primes and write each as prime^exponent.
• Evaluate or simplify using exponent laws: rewrite every term with the same base, apply product/quotient/power rules, then compute.
• Write a large or small number in scientific notation: move decimal so coefficient lies between 1 and 10; the number of moves gives the exponent (positive or negative).
• Count total combinations/passwords: multiply number of choices for each position (often \(10^d\) for \(d\)-digit codes or \(26^6\) for letter locks).
• Compare linear vs exponential growth: model additive steps (linear) versus repeated multiplication (exponential) and contrast magnitudes.
• Estimate real-world large quantities: make reasonable assumptions, express in powers of 10 or scientific notation, then compute.

Common mistakes and exam pitfalls

• Forgetting \(n\neq0\) when using quotient or zero-exponent rules, leading to undefined \(0^0\) or division by zero.
• Sign errors with negative bases: \((-2)^4=16\) but \((-2)^5=-32\); even/odd exponents must be checked.
• Confusing addition of exponents with multiplication of bases (e.g., writing \(2^3\times2^4=4^7\)).
• Placing the decimal incorrectly in scientific notation or forgetting that only the exponent indicates order of magnitude.
• Treating linear growth (additive) as exponential or vice-versa when comparing paper-folding versus step-count problems.
• Missing that \(n^{-a}\) is the reciprocal, not a negative number.
• Overlooking that the exponent, not the coefficient, dominates size in standard form.