Level 4 · Ages 13–15
Read it again any time, including while you work through the exercises.
Multiplying a number by itself over and over is tiring to write: 2 × 2 × 2 × 2 × 2... Powers compress all of that: 2⁵ means 5 factors equal to 2, and is worth 32. In aⁿ, the number a is the base and n is the exponent. Beware of the classic trap: 3⁴ is 3 × 3 × 3 × 3 = 81, and definitely not 3 × 4 = 12!
Powers of 10 are the handiest: 10³ = 1000, 10⁶ = 1000000. The exponent counts the zeros. For tiny numbers, we use negative exponents: 10⁻³ = 0.001, meaning one thousandth. A negative exponent does not make the number negative: it makes it small!
These powers of 10 let us write huge or tiny numbers in scientific notation: a number between 1 and 10 (10 excluded), multiplied by a power of 10. For example 45000 = 4.5 × 10⁴ and 0.00072 = 7.2 × 10⁻⁴. Scientists use it every day: the Earth-Sun distance is written 1.5 × 10⁸ km, much easier to read than 150000000 km.
One last tool: when you multiply two powers of the same base, the exponents add up. 10² × 10³ = 10⁵, because 2 factors of 10 followed by 3 factors of 10 make 5 factors of 10 in total. Makes sense, right?
Compute 2⁵.
2⁵ = 2 × 2 × 2 × 2 × 2
The exponent 5 tells you the number of factors: five 2s multiplied together, not 2 × 5.
2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16
I multiply step by step: 4, then 8, then 16.
16 × 2 = 32
One last factor of 2 and I get 32.
Result: 2⁵ = 32
Write 45000 in scientific notation.
45000 → 4,5
I place the decimal point right after the first nonzero digit: I get 4.5, a number between 1 and 10.
4,5 × 10⁴ = 45000
To go from 4.5 to 45000, the decimal point moves 4 places to the right: the exponent is 4.
Result: 45000 = 4.5 × 10⁴