Math Help

When addition, subtraction, division and exponentiation (raising to a power) appear in an expression and that expression must be simplified, the operations should be performed in the following sequence:

• Simplify all expression within parentheses, brackets, and braces from the inside out.
• Perform all exponential operations, proceeding from left to right.
• Perform all products and quotients, proceeding from left to right.
• Perform all sums and differences, proceeding from left to right.

The following are examples of this process, in which the order of the numerals and operations is the same in each case, but the groupings differ:

[(2 + 3) (-3 -1)²]²
= [5 x (-4)²]²
= (5 x 16)²
= 80²
= 6400

[(2 + 3 x (-3) -1)²]²
= [(2 + (-9) -1)²]²
= (-8²)²
= 64²
= 4096

A note of caution is in order here: This rule doesn’t apply to exponents of exponents. For example, 3³ raised to power of 3 is equal to 27³ or 19,683. But 3 raised to power of 3³ is equal to 3²⁷ or 7,625,597,484,987.

Several properties, also called principles or laws, are recognized as valid for the operations of addition, subtraction, multiplication, and division for all real numbers. Here are some of them. It’s not a bad idea to memorize these. You probably learned them in elementary school:

Additive Identity Element: When 0 is added to any real number a, the sum is always equal to a. The number 0 is said to be the additive identity element: a + 0 = a.

Multiplicative Identity Element: When any real number a is multiplied by 1, the product is always equal to a. The number is said to be the multiplicative identity element: a x 1 = a.

Additive Inverses: For every real number a, there exists a unique real number -a such that the sum of the two is equal to 0. The numbers a and -a are called additive inverse: a + (-a) = 0.

Multiplicative Inverses: For every nonzero real number a, there exists a unique real number 1/a such that the product of the two is equal to 1. The numbers a and 1/a are called multiplicative inverses: a x (1/a) = 1. The multiplicative inverse of a real number is also called its reciprocal.

Commutative Law For addition: When any two real numbers are added together, it does not matter in which order the sum is performed. The operation of addition is said to be commutative over the set of real numbers. For all real numbers a and b, the following equation is valid: a + b = b + a.

Commutative Law For Multiplication: When any two real numbers are multiplied by each other, it does not matter in which order the product is performed. The operation of multiplication, like addition, is commutative over the set of real numbers. For all real numbers a and b, the following equation is always true: a x b = b x a. A product can be written without the “times sign” (x) if, but only if, doing so does not result in an ambiguous or false statement. The above expression is often seen written this way: ab = ba.

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When a real number with a plus sign or a minus sign is raised to a positive integer power n, the following rules apply:

(+)ⁿ = (+)
(-)ⁿ = (-) if n is odd
(-)ⁿ = (+) if n is even

Reciprocal of reciprocal: For all nonzero real numbers, the reciprocal of the reciprocal is equal to the original number. The following equation holds for all real numbers a provided that a ≠ 0: 1/(1/a) = a.

Product of sums: For all real nubmers a, b, c, and d, the product of (a + b) with (c + d) is given by the following formula: (a + b)(c + d) = ac + ad + bc + bd.

Cross multiplication: Given two quotients or ratios expressed as fractions, the numerator of the first gimes the denominator of the second is equal to the denominaotor of the first times the numerator of the second. Mathematically, for all real numbers a, b, c, and d where neither a nor b is equal to 0, the following statement is valid: a/b = c/d | ad = bc.

Reciprocal of product: For any two nonzero real numbers, the reciprocal (or minus-one power) of their product is equal to the product of their reciprocals. If a and b are both nonzero real numbers:

1/(ab) = (1/a)(1/b)
(ab)^-1 = a^-1b^-1

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