Squares, powers and primes: cheat sheet

Number Foundations · The words and methods from the lessons, with an example of each.

A number times itself: the area of a square with that side. The perfect squares to 144: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

6^2 = 6 \times 6 = 36

Find the perfect squares either side and take their roots. Square decimals to get closer: $6.6^2 = 43.56$ , so $\sqrt{43} \approx 6.6$ .

36 < 43 < 49 \;\Rightarrow\; 6 < \sqrt{43} < 7

HCF: the primes both share. LCM: all the primes, shared ones once. Sharing into equal groups needs the HCF; things lining up again need the LCM.

12 = 2^2 \times 3,\; 18 = 2 \times 3^2:\; \text{HCF} = 2 \times 3 = 6,\; \text{LCM} = 2^2 \times 3^2 = 36

The number that times itself makes this one: the side of a square with that area. Not half: $\sqrt{36} \ne 18$ .

\sqrt{36} = 6 \text{ because } 6 \times 6 = 36

The base (2) is multiplied; the exponent (5) counts how many times. $2^5$ is not $2 \times 5$ .

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

A number with exactly two factors, 1 and itself. 1 is not prime.

2,\; 3,\; 5,\; 7,\; 11,\; 13,\; \ldots

Split into factor pairs until every branch is a prime, then write repeated primes with exponents. Every tree gives the same primes.

60 = 2^2 \times 3 \times 5