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(+)(+) = (+)
These rules apply to both multiplication and division.
When multiplying and even number of negative factors, the product is positive.
When multiplying an odd number of negative factors, the product is negative.
b/a is the reciprocal of a/b
The reciprocal is also called the "multiplicative inverse" of a number
invert, or flip, the fraction
To find the reciprocal of a fraction...
-2/5 = -2/5 = 2/-5
A negative fraction can be written in three different, but equivalent ways:
Division
process of breaking a quantity into equal parts
quotient
the answer to a division problem
Dividend
the number that is being divided
divisor
the number that divides another number.
To Divide Signed Numbers
divide the absolute values or numerical parts, and determine the sign of the quotient.
the quotient is positive.
If the dividend and the divisor have the same sign
the quotient is negative.
If the dividend and the divisor have different signs
To divide by a fraction
find the reciprocal of the divisor, find the any common factors, and multiply.
To divide by a mixed number
rewrite as an improper fraction, find the reciprocal of the divisor and multiply.
exponent
A mathematical notation indicating the number of times a quantity is multiplied by itself
The base is the number being multiplied
The exponent is the number of times the base is used as a factor
2 to the fourth power or 2⁴
The exponential form of 2 x 2 x 2 x 2 is
Sign Rule for Exponents
When the base is negative, be especially careful in determining the sign of the answer.
A negative base raised to an even power is positive
A negative base raised to an odd power is negative
X² = squared
X³ = cubed
Exponent Expressions with Negative Signs
Be careful how you read expressions with exponents and negative signs.
(-2)⁴ means (-2)(-2)(-2)(-2) = 16
When the negative sign is inside the parentheses, the negative applies to all the bases
-2⁴ means (-2)(2)(2)(2) = -16
When there are no parentheses, the negative only applies to the first base
Special Cases with Exponents
Any non-zero number raised to the zero power is equal to 1 x⁰ = 1 (b ≠ 0)
The Distributive Property
For all real numbers a, b, and c, a(b + c) = ab + ac
The Distributive Property is most useful when it does not conflict with the order of operations.
That is, it is mostly used when you cannot perform the operations inside the parentheses. The Distributive Property can be used to remove the parentheses.
commutative property of multiplication
If the order of factors changes, the product stays the same.
associative property of multiplication
When the grouping of factors changes, the products stays the same.
identity property of multiplication
The product of 1 and any number is that number.
algebraic expression
An expression consisting of one or more numbers and variables along with one or more arithmetic operations
term
An element or number in a sequence.
Like Terms
expressions that contain the same variables to the same power
Combining like terms
If an expression has several terms, you might find it helpful to rearrange the terms so that like terms are together.
Use the Distributive Property to remove the parentheses.
Many expressions in algebra use grouping symbols such as (parentheses), {braces}, or [brackets].
the opposite of what is inside the parentheses.