More About Whole Numbers

Exponents

       Repeated addition is shortened by using multiplication:

16 + 16 + 16 + 16 + 16=1,048,567.

       Repeated multiplication can be shoretened by using exponents. Thus, if 16 is used as a factor five times, we can write

16 x 16 x 16 x 16 x 16=165 =1,048,567.

       The expression such as 53 is read as "5 cubed," "5 raised to third power," or "the third power of 5." Parts of 53 are given names.

5 is called the base.

3 is called the exponent.

53 is called a power.

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Power, Base, and Exponent

In the expression xy

x is called the base,

y is called the exponent, and

xy is called a power.

Remark:

Don't confuse the words exponent and power. The exponent 3 is just the number that tells how many 5s to multiply. THe power is the entire expression, 53. Power is similar in meaning to sum or product.

5 + 3 -- A sum

5 - 3 -- A difference

5 x 3 -- A product

5 / 3 -- A quotient

53 -- A power

Order of Operations

       When an expression has two or more operations, difficulties involving the order of operations may arise. For instance, what number does 13 + 32 x 7 represent?

       If we add 13 and 32 first and then multiply by 7, we get

45 x 7=315.

       But if we multiply 32 by 7 first, then add the result to 13, we get a different number.

13 + 224=237

       To avoid this difficulty, grouping symbols, such as parentheses ( ), brackets [ ], and braces { } are used to tell which operation to do first.

(13 + 32) x 7 -- means 45 x 7, or 315.

13 + (32 x 7) -- means 13 + 224, or 237.

12 / [2 x 3] -- means 12 / 6, or 2.

       Mathematicians around the world have agreed on definite order for simplifying (or eveluating) any mathematical expression involving grouping symbols, addition, subtraction, multiplication, dividion, and exponents. Under these rules,

5 x 22 + 14 / 2=5 x 4 + 14 / 2 -- First, work out 22=4.

=20 + 14 / 2 -- Then, work out 5 x 4=20.

=20 + 7 -- Divide 14 by 2.

=27 -- Finally, do the addition.

So, the following order of operations is agreed upon.

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Rules for Order of Operations

  1. First, simplify expressions inside grouping symbols, such as parentheses (), brackets [], or braces {}. Start with the innermost grouping.

  2. Second, find any powers indicated by exponents.

  3. After powers, multiply and divide in the order they occur, from left to right.

  4. Last, add and subtract in the order they occur, from left to right.

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The following are different ways of asking the same question.

  • Evaluate.

    • Example:

    Evaluate 30 - {15 + (40 / 5)}.

    Solution:

    30 - {15 + (40 / 5)}

    =30 - {15 + 8} -- First, evaluate (40 / 5) = 8.

    =30 - 23 -- Then, evaluate {15 + 8} = 23.

    = 7 -- Finally, do the subtraction.


  • Simplify.

    • Example:

    Simplify {[85 - (3 + 2)] / 4} + 18 x 2.

    Solution:

    {[85 - (3 + 2)] / 4} + 18 x 2

    ={[85 - 5] / 4} + 18 x 2 -- Evaluate (3 + 2)=5.

    ={80 / 4} + 18 x 2 -- Evaluate [85 - 5]=80.

    =20 + 18 x 2 -- Evaluate {80 / 4}=20.

    =20 + 36 -- Evaluate 18 x 2=36.

    =56 -- Evaluate 20 + 36.


  • Give the simplest answer.

    • Examples:

    24 / [54 - (28 + 18)].

    Solution:

    24 / [54 - (28 + 18)]

    =24 / [54 - 46] -- First, evaluate (28 + 18)= 46.

    =24 - 8 -- Next, evaluate [54 - 46]=8.

    =3 -- Finally, evaluate 24 / 8=3.