Digits, Grade 8, Volume 2, Homework Helper

# Properties

Unless otherwise stated, the variables a, b, c, m, and n used in these properties can be replaced with any number represented on a number line.

Identity Properties

n + 0 = n and 0 + n = n

Multiplication

n · 1 = n and 1 · n = n

Commutative Properties

a + b = b + a

Multiplication a · b = b · a

Associative Properties

(a + b) + c = a + (b + c)

Multiplication

(a · b) · c = a · (b · c)

Inverse Properties

a + (–a) = 0 and −a + a = 0

Multiplication

eh middle dot , 1 over eh , equals 1 and , 1 over eh , middle dot eh equals 1 comma . open , eh not equal to 0 , close

Distributive Properties

a(b + c) = ab + ac (b + c)a = ba + ca

a(bc) = abac (bc)a = baca

Properties of Equality

If a = b,

then a + c = b + c.

Subtraction

If a = b,

then ac = bc.

Multiplication

If a = b,

then a · c = b · c.

Division

If a = b, and c ≠ 0,

then

Substitution

If a = b, then b can replace a in any expression.

Zero Property

a · 0 = 0 and 0 · a = 0.

Properties of Inequality

If a > b,

then a + c > b + c.

If a < b,

then a + c < b + c.

Subtraction

If a > b,

then a − c > b − c.

If a < b,

then a − c < b − c.

Multiplication

If a > b and c > 0, then ac > bc.

If a < b and c > 0, then ac < bc.

If a > b and c < 0, then ac < bc.

If a < b and c < 0, then ac > bc.

Division

If a > b and c > 0, then eh over c , greater than , b over c .

If a < b and c > 0, then eh over c , less than , b over c .

If a > b and c < 0, then eh over c , less than , b over c .

If a < b and c < 0, then eh over c , greater than , b over c .

Properties of Exponents

For any nonzero number n and any integers m and n:

table with 7 rows and 2 columns , row1 column 1 , cap zero , , cap exponent , column 2 a super 0 end super , equals1 , row2 column 1 , cap negehtive , , cap exponent , column 2 eh super negative n end super , equals , fraction 1 , over eh to the n end fraction , row3 column 1 , cap product , of , cap powers , column 2 eh to the m , middle dot , eh to the n , equals . eh super m plus n end super , row4 column 1 , cap power , of eh , cap product , column 2 open eh b close to the n . equals , eh to the n , b to the n , row5 column 1 , cap quotient , of , cap powers , column 2 fraction eh to the m , over eh to the n end fraction . equals . eh super m minus n end super , row6 column 1 , cap power , of eh , cap quotient , column 2 open , eh over b , close to the n . equals . fraction eh to the n , over b to the n end fraction , row7 column 1 , cap power , of eh , cap power , column 2 open , eh to the m , close to the n , equals , eh super m n end super , end table

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