Understanding Exponents

Repeated Multiplication can be written in exponential form.

a*a*a*a*a=a{^{5}}

\left ( 2x \right )\left ( 2x \right )\left ( 2x \right )=\left ( 2x \right )^{3}

 

a^{n}=a*a*a...*a

{n factors}{\rightarrow}

 

                              n=exponent

                             a=base

                      read  a^{n}=a to the n^{th} power

Properties of Exponents

  1.     a^{m}*a^{n}=a^{m+n}

  2.    \frac{a^{m}}{a^{n}}=a^{m-n}

  3.   a^{-n}=\frac{1}{a^{n}}=\left ( \frac{1}{a} \right )^{n}

  4.   a^{0}=1

  5.   \left ( ab \right )^{m}=a^{m}b^{m}

  6.   \left ( a^{m} \right )^{n}=a^{m*n}

  7.   \left ( \frac{a}{b} \right )^{m}=\frac{a^{m}}{b^{m}}

  8.   \left | a^{2} \right |=\left | a \right |^{2}=a^{2}

Examples:   \left ( -2x^{2} \right )^{3}\left ( 4x^{3} \right )^{-1}=\frac{-8x^{6}}{4x^{3}}=-2x^{3}

                            3^{n}*3^{2n}=3^{3n}

                           \left ( 4a^{-2}b^{3} \right )^{-3}=\left ( \frac{1}{4a^{-2}b^{3}} \right )^{3}=\frac{1}{4^{3}a^{-6}b^{9}}=\frac{a^{6}}{64b^{9}}

 

The above examples use properties 1,2,3,5, and 6. 

More Advanced topics related to Exponents

1.  Scientific Notation

To learn about Scientific Notation see the post called Scientific Notation on this website.

2.  Rational Exponents

Rational exponents are fractions. 

Example:         a^{\frac{1}{n}}=\sqrt[n]{a},\, \, where \, \, \frac{1}{n}\, \, is \, \, the\, \, rational\, \, exponent\, \, of \, \, a.

                         a^{\frac{m}{n}}=\sqrt[n]{a^{m}}

                          a^{\frac{m}{n}}=\left ( a^{\frac{1}{n}} \right )^{m}=\left ( \sqrt[n]{a} \right )^{m}=\sqrt[n]{a^{m}}

                        or     a^{\frac{m}{n}}=\left ( a^{m} \right )^{\frac{1}{n}}=\sqrt[n]{a^{m}}

              

                     b^{\frac{m}{n}}=\sqrt[n]{b^{m}}                              n^{th}\, \, root\, \, of\, \, b^{m} 

                                                                             b=base

                                                                            m=power\, \, of\, \, the\, \, base

                                                                            n=index\, \, of\, \, the\, \, \sqrt{\, \, }\, \, or\, \, root 

Keep practicing using the properties of exponents to improve your understanding of exponents. 

Filed under: Pre-requisites for Pre-Calculus

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