Exponential Notation


Exponential notation is a concise way of denoting a number or
expression repeatedly multiplied by itself.

Examples:
(1)   x²  = (x)(x)     (2)  4³  = (4)(4)(4)

In the expression x² the number two is called the exponent,

x is called the base of the exponenential expression.

Note the exponent indicates the number of times the base is multiplied by itself.

Current versions of Hyter Text Markup Language do not support mathematical notations adequately. Only exponents of two are three can be handled by some browsers at this time.

 Warning:  -3²  and (-3)²  do not represent the same number.

-3² = -(3)(3) = -9   while  (-3)²  = (-3)(-3) = 9.

If parenthes are used we multiply what is inside parenthesis
by itself the appropriate number of times.

 Warning:  2y³  and (2y)³  do not represent the same expression.

 2y³ = 2(y)(y)(y)   while   (2y)³ = (2y)(2y)(2y) = 8y³.


Working with Exponents

Consider (x² )(x³ ) = [(x)(x)][(x)(x)(x)] Note if we multiply two factors of x by three factors of x we obtain five factors of x which could be indicated by x raised to the 5th power. The exponent five is obtained by adding the two to three. Rule: When multiplying exponentials with the same base we add the exponents on the factors together to obtain the exponent on the answer. (Hypertext Markup Language will not currently display x raised to the fifth power in the usual form.) Consider (x³ )/(x²) = [(x)(x)(x)]/[(x)(x)] = x Note if we divide three factors of x by two factors of x and reduce we obtain one factor of x. If we subtract the exponent of two from the exponent of three we obrtain the exponent of one. Rather than write x raised to the first power we mearly write x. Rule: When dividing exponentials with the same base we subtract the exponent on the denominator from the exponent on the numerator to obtain the exponent on the answer. (Remark: In certain instances the exponent on the answer can be negative. This poses no problem for ultimately we will define what is meant by a number or variable raised to a negative exponent.) Consider (x² )³ = (x² )(x² )(x² ) = [(x)(x)][(x)(x)][(x)(x)] Note when we raise two factors of x to the third power we end up with six factors of x or x raised to the sixth power. If we multiply the exponent two by the exponent three we obtain the exponent on our answer which is six. Rule: When raising an exponential to a power we multiply the exponents together.

Three Laws of Exponents

  1. When multiplying exponentials with the same base we add the exponents on the factors to obtain the exponent on the answer. Go to multiplying polynomials.

  2. When dividing exponentials with the same base we subtract the exponent on the denominator from the exponent on the numerator to obtain the exponent on the answer. Go to dividing polynomials.

  3. When raising an exponential to a power we multiply the two exponents together to obtain the exponent on the answer.

    See any algebra text for a more in depth discussion of exponents. In particular for a discussion of zero and negative exponents.

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