---
title: "Squares, powers and primes: cheat sheet"
subtitle: "Number Foundations"
---

# Squares, powers and primes

> The words and methods from this unit, with an example of each.

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**square number**: 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.

$$6^2 = 6 \times 6 = 36$$

_A 6 by 6 square covers 36 unit squares_

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

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

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

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

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

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

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

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

_A square of side 6.6 is just over the dashed target of 43_

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

$$60 = 2^2 \times 3 \times 5$$

_A factor tree for 60 ending in the primes 2, 2, 3 and 5_

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**HCF and LCM**: 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.

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

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