Reference / universal

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Every result here is universal — identities, formulas, powers, roots, trigonometry, exponentials and logarithms. Nothing is grouped by class or syllabus; find things by what they are.

Algebraic Identities

The standard expansion and factorisation identities.

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a^{2} - b^{2} = (a + b) (a - b)

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 ab + 2 b c + 2 c a

a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a)

If a+b+c = 0 then a³+b³+c³ = 3abc.

(x + a) (x + b) = x^{2} + (a + b) x + ab

(a + b)^{2} - (a - b)^{2} = 4 ab

(a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})

Number table

Squares, cubes and roots.

n	n²	n³	√n	∛n
1	1	1	1.00000	1.00000
2	4	8	1.41421	1.25992
3	9	27	1.73205	1.44225
4	16	64	2.00000	1.58740
5	25	125	2.23607	1.70998
6	36	216	2.44949	1.81712
7	49	343	2.64575	1.91293
8	64	512	2.82843	2.00000
9	81	729	3.00000	2.08008
10	100	1000	3.16228	2.15443
11	121	1331	3.31662	2.22398
12	144	1728	3.46410	2.28943
13	169	2197	3.60555	2.35133
14	196	2744	3.74166	2.41014
15	225	3375	3.87298	2.46621
16	256	4096	4.00000	2.51984
17	289	4913	4.12311	2.57128
18	324	5832	4.24264	2.62074
19	361	6859	4.35890	2.66840
20	400	8000	4.47214	2.71442

Powers

n¹ to n¹⁰ for n = 1…10.

n	n¹	n²	n³	n⁴	n⁵	n⁶	n⁷	n⁸	n⁹	n¹⁰
1	1	1	1	1	1	1	1	1	1	1
2	2	4	8	16	32	64	128	256	512	1024
3	3	9	27	81	243	729	2187	6561	19683	59049
4	4	16	64	256	1024	4096	16384	65536	262144	1048576
5	5	25	125	625	3125	15625	78125	390625	1953125	9765625
6	6	36	216	1296	7776	46656	279936	1679616	10077696	60466176
7	7	49	343	2401	16807	117649	823543	5764801	40353607	282475249
8	8	64	512	4096	32768	262144	2097152	16777216	134217728	1073741824
9	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
10	10	100	1000	10000	100000	1000000	10000000	100000000	1000000000	10000000000

Standard angles

Exact trigonometric values.

θ	sin θ	cos θ	tan θ
0°	$0$	$1$	$0$
30°	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
45°	$\frac{1}{2}$	$\frac{1}{2}$	$1$
60°	$\frac{3}{2}$	$\frac{1}{2}$	$3$
90°	$1$	$0$	$undefined$
180°	$0$	$- 1$	$0$
270°	$- 1$	$0$	$undefined$
360°	$0$	$1$	$0$