Transcript Simplifying Expressions with Exponents

Simplifying Expressions
with Exponents
Simplify x6 × x5
 The rules tell me to add the exponents. Some students
have trouble keeping the rules straight, so just think
about what exponents mean. The " x6 " means "six
copies of xmultiplied together", and the " x5 " means
"five copies of x multiplied together". So if I multiply
those two expressions together, I will get eleven copies
of x multiplied together. That is:
 X^6 × x^5 = (x^6)(x^5)
= (xxxxxx)(xxxxx) (6 times, and then 5 times)
= xxxxxxxxxxx
(11 times)
= x^11
 Thus:
 X^6 × x^5 = x11
Simplify the following
expression:
 The exponent rules tell me to subtract the
exponents. But let's suppose that I've forgotten
the rules again. The " 6^8 " means I have eight
copies of 6 on top; the " 6^5 " means I have five
copies of 6 underneath.
 How many extra 6's do I have, and where are
they? I have three extra 6's, and they're on top.
Then:
Simplify the following
expression:
 How many extra copies of t do I have, and
where are they? I have two extra copies,
on top:
Simplify the following
expression:
 How many extra copies of 5 do I have, and
where are they? I have six extra copies,
underneath:
Simplify (–46x^2y^3z)^0
 This is simple enough: anything to the
zero power is just 1.
 (–46x^2y^3z)^0 = 1
Simplify the
following expression:
 I can cancel off the common factor of 5 in the
number part of the fraction:
 Now I need to look at each of the variables. How
many extra of each do I have, and where are
they? I have two extra a's on top. I have one
extra b underneath. And I have the same
number ofc's top and bottom, so they cancel off
entirely. This gives me:
Simplifying
Expressions
with
Negative Exponents
Negative Exponents
 Recall that negative exponents mean to move the base
to the other side of the fraction line. For instance:
 In the context of simplifying with exponents, negative
exponents can create extra steps in the simplification
process.
•Simplify the following:
 The negative exponents tell me to move
the bases, so:
 Then I cancel as usual, and get:
Working with Exponents
 When working with exponents, you're dealing
with multiplication. Since order doesn't matter for
multiplication, you will often find that you and a
friend (or you and the teacher) have worked out
the same problem with completely different
steps, but have gotten the same answer in the
end. This is to be expected. As long as you do
each step correctly, you should get the correct
answers. Don't worry if your solution doesn't look
anything like your friend's; as long as you both
got the right answer, you probably both did it
"the right way".
Simplify the following
expression: (–3x^–1y^2)^2
 I can either take care of the squaring
outside, and then simplify inside, or else I
can simplify inside, and then take the
square through. Either way, I'll get the
same answer; to prove this, I'll show both
ways.
simplifying first
squaring first
Simplify the following expression:
(–5x^–2y)(–2x^–3y^2)
 Again, I can work either of two ways: multiply first and
then handle the negative exponents, or else handle the
exponents and then multiply the resulting fractions. I'll
show both ways.
multiplying first/doing the exponents first
 Neither solution method is "better" or "worse" than the
other. The way you work the problem will be a matter of
taste or happenstance, so just do whatever works better
for you.
Simplify the following
expression:
 The negative exponent is only on the x,
not on the 2, so I only move the variable:
Simplify the following
expression:
 The "minus" on the 2 says to move the
variable; the "minus" on the 6 says that
the 6 is negative. Warning: These two
"minus" signs mean entirely different
things, and should not be confused. I have
to move the variable; I should not move
the 6.
Simplifying Expressions
with Exponents:
Complicated
Examples
Simplify the following
expression:
Before I can cancel anything off, I need to simplify
that top parentheses, because it has a negative
exponent on it. I can't cancel off, say, the a's,
because that a4 isn't really on top. I can either
move the whole parentheses down, square, and
then simplify, or I can take the negative-square
through first .
Simplify the following
expression:
 This is a special case. The negative
exponent says that whatever is on top
should go underneath, and whatever is
underneath should go on top. So I'll just
flip the fraction (remembering to change
the power from a negative to a positive),
and simplify from there.
 Warning: This only works if the negative
exponent is on the whole fraction.
Simplify the following
expression:
flip inside, simplify, negative cube, flip, and simplify:
flip inside, simplify, flip the fraction, and cube:
flip the fraction, simplify inside, cube, flip the negative exponents, and simplify:
flip the fraction, flip the negative exponents, simplify, and cube:
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