Video Transcript
Simplify 𝑚 over 𝑛 to the power
negative one all to the power negative three times two 𝑚 to the power negative two
over 𝑛 to the power negative two all to the power negative three.
In order to simplify this
expression, we will need some of the rules of exponents. Because we have fractions raised to
a power, then we can use one of the power laws, which tells us that 𝑎 over 𝑏 to
the power 𝑚 is equal to 𝑎 to the power 𝑚 over 𝑏 to the power 𝑚, where 𝑏 is
nonzero and 𝑚 is in the real numbers. So let’s apply this rule to the
first part of the expression. As we have this fraction to the
power negative three, then we know that this will be equivalent to the numerator to
the power negative three over a denominator to the power negative three. However, to simplify the
denominator of 𝑛 to the power negative one to the power negative three, we’ll need
another rule of exponents.
The rule that we need is one of the
power laws, which tells us that 𝑎 to the power 𝑚 to the power 𝑛 is equal to 𝑎 to
the power of 𝑚 times 𝑛. We take the two exponents of
negative one and negative three and multiply them. And we know that negative one
multiplied by negative three is three. We have now simplified this part of
the expression to 𝑚 to the power negative three over 𝑛 to the power three. Let’s see if we can simplify the
second part of this expression in the same way.
The first thing we can do is apply
this rule for powers of fractions. So the numerator will be equivalent
to two 𝑚 to the power negative two to the power negative three. And the denominator will be 𝑛 to
the power negative two to the power negative three. In order to simplify the numerator
of this fraction, we will need to recall another law of exponents. This power law tells us that 𝑎𝑏
to the power 𝑚 is equal to 𝑎 to the power 𝑚 times 𝑏 to the power 𝑚, where 𝑚 is
in the set of real numbers. The numerator of this fraction will
therefore simplify to two to the power of negative three times 𝑚 to the power of
negative two to the power of negative three. We can also simplify the
denominator, remembering that we can use this second power law here to multiply the
exponents. And negative two times negative
three gives us six, so the denominator will become 𝑛 to the power six.
The next stage in our working will
be to simplify this part of the expression, 𝑚 to the power of negative two to the
power negative three. Just as before, we can multiply
these exponents. So we have 𝑚 to the power of
negative two times negative three. And we know that negative two times
negative three is six. Now, we could potentially simplify
this expression a little further by dealing with the two to the power of negative
three. But for now, let’s substitute these
values in orange and pink for the parts of the expression. When we multiply these together, we
have 𝑚 to the power negative three over 𝑛 to the power three times two to the
power negative three 𝑚 to the power six over 𝑛 to the power six.
We know that when we multiply
fractions, we multiply the numerators and multiply the denominators. We might then notice that on the
numerator, we have two values of the same base of 𝑚. And there is an exponent rule to
help us work this out. This rule tells us that 𝑎 to the
power 𝑚 times 𝑎 to the power 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. Of course, the values that we’re
using in the question of 𝑚 and 𝑛 are not the same values that we’re using in these
exponent rules. And so on the numerator, we will
add the two exponents for 𝑚 of negative three and six. So we have two to the power
negative three times 𝑚 to the power of negative three plus six on the
numerator. On the denominator, we add the
exponents three and six of 𝑛. So we have 𝑛 to the power of three
plus six.
At this point, we have simplified
the variables of 𝑚 and 𝑛 as much as we can. But let’s see if we can do anything
to simplify this two to the power of negative three. And we can use one final exponent
rule for negative indices. This rule tells us that 𝑎 to the
power of negative 𝑛 is equal to one over 𝑎 to the power 𝑛. This means that two to the power of
negative three can be written as one over two cubed. We should remember that two cubed
is equal to two times two times two, and that’s eight. So two to the power of negative
three is equal to one over eight. And when we plug in one-eighth in
place of two to the power negative three, we have the expression 𝑚 cubed over eight
𝑛 to the power nine. And this is the answer. We have simplified the given
expression as much as we can to give 𝑚 cubed over eight 𝑛 to the power nine.
