hi welcome to moomoomath today we're gonna talk about properties of exponents
the first property is called the product of powers property we've got a to the M
times a to the N is equal to the a the M plus n so we're adding our exponents
when we multiply basis the second property is called the power of powers
property we've got a to the n all raised to the nth power that is equal to a to
the M times N and we're gonna multiply the exponents the third one is the power
of a product property that's when you have the product all quantity to the
M power we can rewrite that as a to the M times B to the M this fourth property
would be called the zero exponent property that's when a base a is raised
to the zero power that is always equal to one as long as a is not zero the
fifth one is the quotient of powers property we've got a over M divided by a
to the n that's equal to the a to the M minus n in other words we subtract our
exponents when you divide like bases Alice run through a couple examples
we're gonna start with a product of powers property the first example is 4
to the second times 4 to the fourth well 4 is our like base so that's gonna be 4
to the 2 plus 4 or 4 to the sixth power a second example of that product of
powers would be a to the six times a to the third we since we're multiplying
like bases we're gonna add the exponents that will give us a to the ninth power
the second property is the power of a power the first one we're gonna look at
is X to the fifth all raised to the second power well what we do there is we
multiply the exponents so that's gonna give us X to the tenth power a second
example of that one would be five to the negative second all raised to the third
although the exponent is negative we still rate multiply those two exponents
together and that gives us 5 to the negative six power the third property is
the product power of a product property the first one we're gonna look at is a
to the second I'm sorry four times a squared b cubed that quantity all raised
to the fifth power or what we're gonna do is take that five and distribute it
to every element in the parenthesis so that will give us a four for the fifth
power a squared to the fifth power which we
know we multiply those exponents so that results in a to the tenth and then B to
the third all to the fifth we know we multiply the exponent so that becomes B
to the fifteenth you can use your calculator if you want to raise four to
the fifth power and get the coefficient or you can just leave it as four to the
fifth as the coefficient a second example of a power of a product property
would be X to the fourth Y to the seventh all raised to the third power we
would then distribute that 3 to both exponents
and we're raising to a power we're multiplying exponents so that will be X
to the 12 I'm seven I'm sorry why there's 21st that's the example the zero
exponent property no matter what you are raising to the zero power it's always
gonna be one so the example we're going to look at is X cubed all to the zero
power well when we multiply exponents that's going to give us X to the zero
and anything to the zero power is one so X to the third to the zero becomes one
and the final property is the quotient of powers property so we're looking at
division quotient always means to divide let's start with X to the eighth divided
by X to the sixth we subtract our exponents so it's going to be X to the
eighth minus 6 or x squared is your final answer the second example would be
X to the fifth on top divided by X to the eighth on the bottom we still
subtract the exponents so it will be X to the 5 minus 8 which is X to the
negative third you can also rewrite that as 1 over X cubed
and those are the five properties of exponents I hope this video has been
helpful
you