We want to simplify the given expressions.
An expression is not considered simplified
if it contains negative exponents.
Looking at this first expression,
we want to begin by simplifying inside the parentheses.
So looking at our fraction,
notice how we have a numerator of two to the first
times b to the first.
Our denominator is b to the third,
but we can think of this as one b to the third.
So when we simplify the fraction,
notice two divided by one would be two,
and then we have b to the first divided by b to the third,
because we're dividing and the bases are the same
we subtract the exponents.
So we have b raised to the power of one minus three.
And all this is still raised to the power of negative two.
Well one minus three is equal to negative two,
so we have two b to the negative two,
raised to the power of negative two.
Again this is two to the first.
So we have powers to powers,
so we multiply the exponents.
So here we'll have two raised to the power of
one times negative two,
that's negative two.
B to the power of negative two times negative two
is positive four.
And now to change the sign of the exponent on two,
let's write this as a fraction.
If we move two to the negative two to the denominator,
it's going to change the sign of the exponent.
So this is going to be equal to
b to the fourth in the numerator.
And the denominator is now going to be
two raised to the power of positive two or two squared,
which is equal to four.
So our expression simplifies to b to the fourth
divided by four.
Let's look at our next expression.
And again if it's helpful,
we can think of the denominator as having
a coefficient of one.
So we'll simplify one step at a time.
Notice, five divided by one is equal to five.
And we have v to the negative four divided by
v to the negative six.
We're dividing and the bases are the same
so we subtract the exponents,
so we'd have v raised to the power of negative four
minus negative six.
And then we have m to the fifth divided by m to the seventh,
so we'll subtract the exponents.
M to the power of five minus seven.
Simplifying we have five times v raised to the power of,
this becomes negative four plus six
which is positive two.
And then we have m raised to the power of five
minus seven that's negative two.
So we can't have m to the power of negative two
in our simplified expression.
So let's create a fraction with a denominator of one.
And now we'll move m to the power of negative two
down to the denominator,
which changes the sign of the exponent.
So we'll have a numerator of five v squared,
a denominator of m raised to the power of positive two,
or m squared.
This would be our simplified expression.
Next we have 1/4 times 5/6 raised to the power
of negative two.
Now we can use this property here,
but I prefer to think of this as 5 to the first divided by
six to the first.
And because we have powers to powers,
we can multiply the exponents.
So this is equal to 1/4 times
five to the power of negative two
divided by six to the power of negative two.
And now to make the exponents positive,
we'll move five to the power of negative two
down to the denominator,
and six to the power of negative two
up to the numerator.
So this would give us 1/4 times 6 to the positive two power,
or six squared,
divided by five to the power of positive two,
or five squared.
Which is equal to 1/4 times,
here we have 36 divided by 25.
Here we do have a common factor of four.
There's one four in four,
and nine fours in 36.
So for our simplified fraction,
our numerator is going to be one times nine, or nine.
And the denominator is going to be one times 25, or 25.
Now our last expression looks like a big mess,
but notice how this entire base is being raised
to the zero power.
Anything raised to the zero power,
other than zero to the zero,
is equal to one.
And therefore, this equals one.
There's no need to simplify inside the parentheses.
I hope you found this helpful.