Concepts: Power Play (part 1)

1.1 A sheet of paper that reaches the Moon

Take any sheet of paper and fold it in half. Fold again, and again. The thickness doubles with every fold. If the sheet starts at 0.001 cm, then after one fold it is 0.002 cm, after two folds 0.004 cm, after three 0.008 cm, and so on. The table in the textbook lists the values: after 10 folds the thickness is 1.024 cm, after 17 folds about 131 cm, after 26 folds about 670 m, and after 30 folds roughly 10.7 km, higher than Mount Everest.

After forty-six folds the stack would stand over 7,00,000 km tall. The Moon is about 3,84,400 km away, so the folded paper passes it. That jump shows what multiplicative, or exponential, growth can do. The number of folds is small, but the repeated doubling produces enormous numbers.

1.2 Exponential notation

Repeated multiplication is written with a base and an exponent. If n is multiplied by itself a times, the short form is n^a. For example, 5 x 5 x 5 x 5 = 5^4 = 625. Here 5 is the base and 4 is the exponent, also called the power. The expression 5^4 is the exponential form of 625. The powers of 5 are 5^1, 5^2, 5^3, 5^4 and so on.

Two rules of reading help. The expression n^2 is said as n squared; another reading is "n to the power two". The expression n^3 is said as n cubed, or "n to the power three". Any natural number can be an exponent, so n^7 means n multiplied by itself seven times.

Multiplying by 2 ten times gives 2^10 = 1024. This explains why the folded paper is just over 1 cm thick after ten folds: 0.001 cm x 1024 = 1.024 cm. The same 1024 appears whenever ten doublings are involved.

1.3 More examples

Exponential form works for numbers other than 2. The number 4 x 4 x 4 = 4^3 = 64, and (-4) x (-4) x (-4) = (-4)^3 = -64. A negative base gives a positive result when the exponent is even and a negative result when the exponent is odd.

When several letter-numbers are multiplied, like a x a x a x b x b, we group the repeats and write a^3 b^2, read as "a cubed b squared". Similarly, a x a x b x b x b x b becomes a^2 b^4.

It is worth keeping addition and multiplication apart. The sum 4 + 4 + 4 = 3 x 4 = 12, while the product 4 x 4 x 4 = 4^3 = 64. A small change in the operation gives a very different result.

1.4 Prime factors in exponential form

Every composite number can be broken into prime factors and written with exponents. For 32400, divide repeatedly by the smallest possible primes: 32400 = 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5. Grouping the repeats gives 2^4 x 3^4 x 5^2. This compact form is often easier to read than a long product.

The same method prepares the ground for the rules of exponents, because the rules only collect repeated factors of the same base.

1.5 Worked examples

Example 1. Write 648 as a product of powers of primes. Divide by 2: 648 = 8 x 81 = 2^3 x 3^4. Example 2. Find the value of 7^2 x 2^3. Compute 7^2 = 49 and 2^3 = 8, so the product is 392. Example 3. What is 2 x 10^3? Since 10^3 = 1000, the value is 2000. Example 4. Write 5 x 5 x 7 x 7 x 7 in exponential form. The base 5 repeats twice and the base 7 repeats three times, so the answer is 5^2 x 7^3.

1.6 Why exponential growth surprises us

Our intuition is trained on addition, not multiplication. Adding the same number many times gives a steady, predictable climb. Multiplying by the same factor starts slowly and then explodes. Ten doublings take 0.001 cm to about 1 cm, twenty doublings take it to about 1 m, and forty doublings take it to about 1000 km. Each block of ten doublings multiplies the previous answer by roughly 1024.

This is why a small number of folds can cross huge distances while a ladder needs an enormous number of steps. The two processes start from similar places but follow completely different rules of growth.

1.7 Products of letters and numbers

Exponential notation works for letters as well as numbers. Letters stand for unknown or variable numbers, so a x a x a means a^3 and p x p x p x p x p x p means p^6. A product such as 2 x 2 x a x a has two factors of 2 and two factors of a, so it becomes 2^2 a^2.

When two different letters appear with different repeat counts, write one power for each letter: a x a x b x b x b x b becomes a^2 b^4. The same habit is used later for the laws of exponents, where powers with the same base are combined.

1.8 Reading an exponent carefully

The number 5^4 is not 5 x 4. The exponent counts factors, not a multiplier. A useful check is to expand a power into factors and count them. Both 5^4 = 625 and 4^5 = 1024 show that the base matters as much as the exponent. For a quick order-of-magnitude check, remember that powers of 2 and powers of 10 are close: 2^10 = 1024 is just above 10^3 = 1000.