Exponential Properties Involving Products

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How to Use Basic Index Laws

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Key Questions

  • Exponential notation is a way of shorthand for very large numbers and very small numbers.

    But first exponents. They are the numbers that you see at the top right of another number, called the base, as in #10^2#, where the 10 is the base and the 2 is the exponent.

    The exponent tells you how many times you multiply the base with itself: #10^2=10*10=100#

    This goes for any number:
    #2^4=2*2*2*2=16#
    #10^5=10*10*10*10*10=100000#

    So #10^5# is a short way of writing a #1# with #5# zeroes! This will come in handy if we deal with really large numbers:

    Example: The distance to the sun is about 150 million kilometers, or 150 billion meters:

    #"150 000 000 000 m"#

    It would be easy to type a zero more or less by mistake, but we can count the zeroes and say the distance is:

    #"15 + 10 zeroes"# = #"15 x 10"^10 "m"#

    Usually this is done so that the first number is between 1 and 9, so the official scientific notation would be

    #"1.5 x 10"^11 "m"#. Agree?

    The exponent will give a good impression of the order of magnitude.

    Exponential or scientific notation can also be used for very small numbers, such as the mass of an electron, which is #"9.11 x 10"^(-31) "kg"# to three significant figures. Expanded, this would involve moving the decimal to the left 31 places:

    0.000 000 000 000 000 000 000 000 000 000 911 kg.

  • The product of powers rule states that when multiplying two powers with the same base, add the exponents.

    Examples:

    #a^m*##a^n# = #a^(m+n)#

    #5^2*##5^4# = #5^(2+4)# = #5^6#

    #y^6*##y^2# = #y^(6+2)# = #y^8#

    #2^3*##2^5*##2^6# = #2^(3+5+6)# = #2^14#

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