Reference · module 4
Powers and roots
Everything this module explains, on one page.
Powers
Read powers and multiply and divide them quickly.
- A power is repeated multiplication: 2⁵ means 2 × 2 × 2 × 2 × 2, which is 32. The small number, the exponent, counts how many times the base is multiplied.
- Multiplying powers of the same base adds the exponents: a³ × a⁴ is a⁷, because three a's and four a's make seven. Dividing subtracts them: a⁶ ÷ a² is a⁴.
- Any non-zero number to the power 0 is 1: 5⁰ is 1. It follows from division: a³ ÷ a³ is a⁰, and anything divided by itself is 1.
The mistake you are about to make
2⁵ = 10
2⁵ = 32
The 2 and 5 are side by side like in multiplication, so 2 × 5 comes to mind. The exponent counts factors: five twos multiplied together.
Square roots
Find square roots and estimate the ones that are not whole.
- Squaring a number multiplies it by itself: 7² is 49. The square numbers worth knowing by heart are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
- The square root undoes squaring: √49 is 7, because 7² is 49. √ always means the non-negative root, even though (−7)² is also 49.
- Most roots are not whole, but you can place them between two whole numbers. 30 lies between 25 and 36, so √30 is between 5 and 6, a bit less than 5.5.
The mistake you are about to make
√16 = 8
√16 = 4
Halving feels like undoing, and 8 is half of 16. The root asks which number times itself gives 16, and that is 4.
Very big and very small
Write huge and tiny numbers in standard form.
- Powers of ten count zeros: 10³ is 1000 and 10⁶ is a million. Multiplying by 10ⁿ moves the decimal point n places to the right.
- Standard form writes a number as one digit before the point times a power of ten. 45 000 is 4.5 × 10⁴, and the Earth-Sun distance is about 1.5 × 10⁸ km.
- A negative exponent means dividing: 10⁻² is 1/100, which is 0.01. So 0.003 is 3 × 10⁻³ — the point moves three places to the left.
The mistake you are about to make
10⁻² = −100
10⁻² = 0.01
A minus sign usually makes a number negative, so −100 seems natural. In an exponent the minus means dividing: one hundredth.