Outro W8

6.5 Divide Monomials

• Quotient Property for Exponents:
  • If a is a real number, a≠0, and m,n are whole numbers, then:
    $\frac{a^m}{a^n}=a^{m-n}$, m>n     and     $\frac{a^m}{a^n}=\frac{1}{a^{m-n}}$   n>m

• Zero Exponent
  • If a is a non-zero number, then $a^0=1$
    
    
• Quotient to a Power Property for Exponents:
  • If and b are real numbers, b≠0$,$ and m is a counting number, then:
    $\left(\frac{a}{b}\right)^m=\frac{a^m}{a^m}$
  • To raise a fraction to a power, raise the numerator and denominator to that power.
    
    
• Summary of Exponent Properties
  • If a,b are real numbers and m,n are whole numbers, then
    Product Property                         \(a^m\cdot a^n=a^{m+n}\)

           Power Property                            \(\left(a^m\right)^n=a^{mn}\)

           Product to a Power                       \(\left(ab\right)^m=a^m\cdot b^m\)

           Quotient Property                           \(\frac{a^m}{a^n}=a^{m-n}\)    a≠0,    m>n

                                                                  \(\frac{a^m}{a^n}=\frac{1}{a^{n-m}}\)     a≠0,   n>m

           Zero Exponent Definition                \(a^0=1\)     a≠0,

           Quotient to a Power                         \(\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}\)          b≠0

6.6 Divide Polynomials

• Fraction Addition
  • If a,b,andc are numbers where c≠0, then
    $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$      and    $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
    
• Division of a Polynomial by a Monomial
  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

6.7 Integer Exponents and Scientific Notation

• Property of Negative Exponents
  • If n is a positive integer and a≠0, then $\frac{1}{a^{-n}}=a^n$
• Quotient to a Negative Exponent
  • If a,b are real numbers, b≠0 and n is an integer , then $\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n$
• To convert a decimal to scientific notation:
  1. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
  2. Step 2. Count the number of decimal places, n$n$, that the decimal point was moved.
  3. Step 3. Write the number as a product with a power of 10. If the original number is:
     • greater than 1, the power of 10 will be $10^n$
     • between 0 and 1, the power of 10 will be $10^{-n}$
  4. Step 4. Check.
     
     
• To convert scientific notation to decimal form:
  1. Step 1. Determine the exponent, n, on the factor 10.
  2. Step 2. Move the decimal n places, adding zeros if needed.
     • If the exponent is positive, move the decimal point n places to the right.
     • If the exponent is negative, move the decimal point |n| places to the left.