Algebra rules and explanations

When I was first learning about mathematics, I was given a table to learn. I found it difficult to learn why one thing is equal to another because I could not see the link between the two equations. I worked the links out myself and I have uploaded it here to help you remember these rules but even if you forget, you should be able to work them out when you need to like I do. I insert integer numbers (commonly 2, 3 and 4) to try to figure out the rule. What's true for integers is true for floating point numbers such as 2.7 for the equations below (but not for any equation) so in an exam, you could do a little rough work with integer numbers in order to work out the rule if you have forgotten it or need to clarify that you are correct.

Rule

Reason

n^a • n^b= n^a+b

n² • n³= nn • nnn = nnnnn = n⁵

aⁿ • bⁿ= (ab)ⁿ

a² • b²= aa • bb = (ab)•(ab) = (ab)²

(n^b)^c = n^b+c

(n³)² = (nnn)² = (nnn)•(nnn) = (n⁶)

1	= a^-b
a

1	×	a^-b
a	×	a^-b

Since a^-b/a^-b = 1 then anything multiplied by one make no difference to the value. Multiply the terms to get a^-b

(	a	)	n	=	aⁿ
	b				bⁿ

(	a	)	2	=	aa	=	a²
	b				bb		b²

a - b = c
b - a = -c

This is because you can multiply both sides by -1
(-1)(a - b) = (-1)c
-a + b = -c
b - a = -c

-1	=	1
a - b	=	b - a

-1	×	-1
a - b	×	-1

again, since n/n = 1 then anything multiplied by one make no difference to the value. Now multiply out the terms

n(a + b) = an + bn

2(a + b) = (a + b) + (a + b) = a + a + b + b = 2a + 2b

log(ab) = log(a) + log(b)

This rule is used below

log(aⁿ) = n•log(a)

log(a²) = log(aa) = log(a) + log(a) = 2•log(a)

Removing brackets may be confusing when there are negative signs so I use (-1) and make the equation an addition:
a - (b + c) = a + (-1)(b + c) = a + (-b -c) = a - b - c

Have you found an error or do you want to add more information to these pages?
You can contact me at the bottom of the home page.

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