Methods and worked examples: Power Play

2.1 Laws of exponents

The laws of exponents come from writing out the factors. If the same base is multiplied with different exponents, add the exponents: n^a x n^b = n^{a+b}. For example, 3^4 x 3^3 = 3^7 = 2187, because three factors of 3 and four factors of 3 together make seven factors.

If a power is raised to a power, multiply the exponents: (n^a)^b = n^{ab}. For example, (4^3)^2 = 4^6 = 4096. Since multiplication is commutative, (n^a)^b also equals (n^b)^a.

If powers with the same base are divided, subtract the exponents: n^a / n^b = n^{a-b}, where n is not zero. For example, 2^10 / 2^7 = 2^3 = 8.

When a product or quotient is raised to a power, the exponent applies to each factor: n^a x m^a = (nm)^a and n^a / m^a = (n/m)^a. For example, 2^5 x 5^5 = 10^5 = 1,00,000.

2.2 Zero and negative exponents

The pattern of subtracting exponents leads to 2^0. Since 2^4 / 2^4 = 2^0 and any non-zero number divided by itself is 1, we define 2^0 = 1. In general, n^0 = 1 for every non-zero n.

Continuing the pattern gives negative exponents. Halving a line of length 2^4 three times gives 2^1 = 2. Halving it four times gives 2^0 = 1, and halving it five times gives 2^{-1} = 1/2. In general, n^{-a} = 1/n^a, and n^a = 1/n^{-a}. For example, 10^{-3} = 1/1000 and 4^{-2} = 1/16.

2.3 Growing faster and faster

The magical pond is a classic example. A lotus doubles every day, and after 30 days the pond is fully covered. So it was half covered on day 29, and the number of lotuses follows 2^30. A second pond triples daily; four days of tripling multiply the count by 3^4.

The question of the stones follows the same idea. Three daughters each get three baskets; each basket holds three keys; each key opens three rooms; each room has three tables; each table holds three necklaces; each necklace has three diamonds. The total is 3 x 3 x 3 x 3 x 3 x 3 x 3 = 3^7 = 2187. If the counting stops at rooms, there are 3^4 = 81 rooms; stopping at tables gives 3^5 = 243 tables, and stopping at necklaces gives 3^6 = 729 necklaces.

Combinations also multiply. Four dresses and three caps give 4 x 3 = 12 outfits, and seven dresses, two hats and three pairs of shoes give 7 x 2 x 3 = 42 outfits. A 5-digit lock with digits 0 to 9 has 10^5 = 1,00,000 codes, while a 6-slot lock using A to Z has 26^6 codes.

2.4 Powers of 10

Place value is built on powers of 10. The number 47561 is (4 x 10^4) + (7 x 10^3) + (5 x 10^2) + (6 x 10^1) + 1. A decimal such as 561.903 continues the same pattern with negative exponents: (5 x 10^2) + (6 x 10^1) + 1 + (9 x 10^{-1}) + (0 x 10^{-2}) + (3 x 10^{-3}).

Scientific notation writes a number as x x 10^y, where x is at least 1 and less than 10 and y is an integer. For example, 5900 = 5.9 x 10^3, 20800 = 2.08 x 10^4, and 80,00,000 = 8 x 10^6. The exponent matters most because it fixes the size of the number.

2.5 Linear and exponential growth

A ladder with steps 20 cm apart grows linearly: each step adds the same distance. The distance to the Moon needs about 19,22,00,000 steps. A folded sheet reaches the Moon in 46 folds because its thickness multiplies. Linear growth is additive; exponential growth is multiplicative.

The same idea describes very large quantities. A lakh is 10^5, a crore is 10^7, an arab is 10^9; in the international system a million is 10^6 and a billion is 10^9. The number 10^100 is a googol, and 10^googol is a googolplex. Indian and Jaina traditions gave names to very high powers of ten; the Lalitavistara lists names up to 10^53, and a Jaina text reaches 10^96.

2.6 Estimating unknown quantities

Some questions have no exact data, so we guess first and then calculate with reasonable assumptions. For example, to estimate the worth of jaggery equal to a person's weight, assume a weight of 45 kg and a price of 70 rupees per kg; the worth is 45 x 70 = 3150 rupees. Such estimates are not guesses in the dark: they combine a sensible model with the arithmetic of powers.

Worked examples. Simplify 2^5 x 2^3. Add exponents: 2^8 = 256. Simplify (5^2)^3. Multiply exponents: 5^6 = 15625. Find 10^4 / 10^6. Subtract exponents: 10^{-2} = 1/100. Write 0.00032 in scientific notation. Move the decimal point five places: 3.2 x 10^{-4}.

2.7 Comparing and converting scientific forms

To convert a number to scientific notation, place the decimal point after the first non-zero digit and count how many places it moved. Moving left gives a positive exponent; moving right gives a negative one. For example, 65,950 becomes 6.595 x 10^4 because the point moves four places left, and 0.00045 becomes 4.5 x 10^{-4} because the point moves four places right.

To compare two numbers in scientific notation, compare the exponents first. The larger exponent belongs to the larger number, unless the exponents are equal; then compare the coefficients. So 1.496 x 10^{11} is smaller than 1.4335 x 10^{12} because 11 is less than 12.

2.8 Reading very large numbers

Large number names follow powers of ten. One lakh is 10^5, one crore is 10^7, one arab is 10^9, one kharab is 10^{11}, one neel is 10^{13} and one padma is 10^{15}. In the international system one million is 10^6, one billion is 10^9, one trillion is 10^{12} and one quadrillion is 10^{15}. The modern names and the Indian names describe the same numbers with different groupings.

The number 10^100 is called a googol, and 10 raised to a googol is a googolplex. Large as these are, scientific notation handles them with two small symbols: a coefficient and an exponent.