In this video, we’re going to look
at simplifying expressions, particularly expressions that have negative and
fractional exponents in them.
First, we can address what
simplifying an expression is. By simplifying, we mean writing an
expression in the most compact or efficient manner without changing the value of
that expression. The process in simplifying involves
removing parentheses by multiplying factors, combining like terms, or, like we’ll be
considering today, simplifying exponents. We might also wonder why we would
do this simplification process.
Simplifying makes expressions
easier to read and understand. Additionally, it reduces possible
errors in calculating. This is true because after you’ve
simplified an expression, there should be less calculations to do than in an
unsimplified form. To do this simplifying when our
expression has exponents, we’ll need to remember some exponent rules.
We have the rule for multiplying
exponents together. 𝑥 to the 𝑎 power times 𝑥 to the
𝑏 power equals 𝑥 to the 𝑎 plus 𝑏 power. If we have 𝑥 squared times 𝑥
cubed, that will be 𝑥 to the two plus three power, which is 𝑥 to the fifth. Next, we have our quotient
rule. 𝑥 to the 𝑎 power divided by 𝑥 to
the 𝑏 power equals 𝑥 to the 𝑎 minus 𝑏 power. Here, we have 𝑥 to the fifth power
divided by 𝑥 squared, which will be equal to 𝑥 to the five minus two power, 𝑥
The next rule is called a power of
a power. 𝑥 to the 𝑎 power to the 𝑏 power
is equal to 𝑥 to the 𝑎 times 𝑏 power. If we have 𝑥 cubed and we’re
taking that to the fourth power, it will be 𝑥 to the three times four power, 𝑥 to
the 12th power. Another rule, 𝑥 divided by 𝑦 to
the 𝑎 power is equal to 𝑥 to the 𝑎 power over 𝑦 to the 𝑎 power, which is a very
similar rule to 𝑥 times 𝑦 to the 𝑎 power is equal to 𝑥 to the 𝑎 power times 𝑦
to the 𝑎 power.
And then, we have our negative
exponent rule. 𝑥 to the negative 𝑎 power is
equal to one over 𝑥 to the positive 𝑎 power. 𝑥 to the negative fifth power is
equal to one over 𝑥 to the fifth power. By the same rule, we can say one
over 𝑥 to the negative three power is equal to 𝑥 to the positive three power. And that means when we have an
exponent with a negative in the numerator and an exponent with a negative in the
denominator. The 𝑥 to the negative four power
moves into the denominator and becomes 𝑥 to the positive four power. The 𝑦 to the negative three power
in the denominator moves to the numerator and becomes positive.
And before we move on, we should
note that all five of the rules above this are also true when the exponent value is
negative. For example, if we have 𝑥 to the
negative three power being multiplied by 𝑥 to the negative five power, it will be
equal to 𝑥 to the negative three plus negative five power, 𝑥 to the negative
We’re nearly ready to look at some
examples, but first we need to consider fractional exponents. Fractional exponents, sometimes
called rational exponents, represent certain roots. What we’re saying here is that 𝑛th
root of 𝑥 is equal to 𝑥 to the one over 𝑛 power. By this rule, the square root of 𝑥
is equal to 𝑥 to the one-half power. We know that this is the square
because when this radical symbol does not have a value in front of it, it’s the
On the other hand, the cube root of
𝑥 would be equal to 𝑥 to the one-third power. This fractional exponent occurs in
one more form. And that’s when we have 𝑥 to the
𝑚 power and we’re taking the 𝑛th root of that. That will be equal to 𝑥 to the 𝑚
over 𝑛 power. If we have the cube root of 𝑥
squared, that is 𝑥 to the two-third power. The denominator of the exponent
comes from the root, and the numerator comes from the power.
Another thing we need to say about
these kinds of fractional exponents is that it’s also true if you first take the
cube root of 𝑥 and then you square that value. The way that it’s written here, you
would take 𝑥 squared first, and then you would do the cube root. In this format, you would take the
cube root and then you would square that value. Both of these would give you the
same numerical result. And they’re both equal to 𝑥 to the
two-thirds power. Now, we’re ready to consider some
Which of the following is equal to
three-fifths to the negative six power times three-fifths to the negative three
power all over three-fifths to the eight power? (A) Three-fifths to the 11th power,
(B) three-fifths to the negative one power, (C) three-fifths to the negative 11
power, (D) three-fifths to the negative 17th power, or (E) three-fifths to the
negative 25 power.
Here’s our expression. In our numerator, we have two
exponents with the same base being multiplied together. And we know when that happens, we
can simplify by adding the two powers together. This is true even though both of
these values are negative. To simplify then, we’ll have
three-fifths to the negative six plus negative three power, which is three-fifths to
the negative ninth power. We bring across the
denominator. And then we see we’re dividing an
exponent by another exponent with the same base.
And to do that, we know that we can
subtract the power values. This is true even though one of the
powers is negative. It would be three-fifths to the
negative nine minus eight power, three-fifths to the negative 17th power. Now, it’s possible to distribute
this negative 17 to simplify even further. However, our question is just
asking to find an equivalent expression, which we have, three-fifths to the negative
17th power, which is the final answer.
In our next example, we’ll need to
simplify with these fractional powers.
Simplify 16 to the five-fourth
power over 16 to the one-half power.
We have a few choices about the
order in which we choose to solve this. Because both of these values have
the same base, we could subtract their powers, which would make it 16 to the
five-fourth minus one-half power. Now, one-half is two-fourths, which
means we would have five-fourths minus two-fourths, which is 16 to the three-fourth
power. This fractional power tells us
we’re taking the fourth root of 16 cubed.
Because of our power of a power
rule, we can say 16th to the one-fourth power to the third power or we could say 16
cubed to the one-fourth power. 16 to the one-fourth power cubed
would be my preferred option because 16 to the one-fourth power is two, whereas 16
cubed is 4096. From there, two cubed equals
eight. And to take the fourth root of
4096, you would need a calculator. But once you put it in the
calculator, it would tell you that the fourth root of 4096 is eight.
By taking the fourth root first and
then cubing the answer, we were able to keep the values bit smaller than if you did
it the other way around. But both ways show us that this
expression is equal to eight.
Here’s our next example.
Calculate the square root of
one-fourth to the fifth power times one-fourth squared.
Looking at this expression, we
notice that inside the radical, we’re dealing with exponents that have the same
base. We can combine these values by
saying one-fourth to the five plus two power, one-fourth to the seventh power. And then, we can take this radical,
and we can rewrite it as a fractional exponent so that we have one-fourth to the
seven-halves power. And then, we have a choice. We can either take one-fourth to
the one-half power and then take that value to the seventh power. Or we can say one-fourth to the
seventh power, find out what that is, and then take the square root of that
Taking the square root first
usually means the numbers that you’re working with are a bit simpler. We can distribute that one-half
power over the one and the four. The square root of one is one, and
the square root of four is two. So, we now have one-half to the
seventh power. When we distribute that seventh
power across the one and the two, one to the seventh power is one, and two to the
seventh power is 128. The value of this expression is
then one out of 128.
Here’s another example.
Which of the following expressions
has the same value as negative two to the fifth power to the negative third
power? (A) Negative two squared, (B)
negative two to the eighth power, (C) negative two to the 15th power, (D) negative
one over two to the eighth power, or (E) negative one over two to the 15th
We have the expression negative two
to the fifth power to the negative three power. Since we’re taking a power of a
power, we can go ahead and multiply these two indexes together. That would be five times negative
three, which is negative 15. Be careful here; one of the answer
choices is negative two to the positive 15th power. And that’s not what we see
here. So, we might think 𝑥 to the
negative 𝑎 power equals one over 𝑥 to the 𝑎 power, which gives us one over
negative two to the 15th power, but still not exactly what we’re looking for.
One thing we can say about negative
two is that it has factors negative one and two. And that means if we break these
factors apart, we’re able to distribute that power 15. This gives us one over negative one
to the 15th power times two to the 15th power. Because 15 is an odd number,
negative one to the 15th power is negative one. We know this because negative one
times negative one is positive one, but negative one times negative one times
negative one is negative one.
We take that multiplied by negative
one and make this fraction negative so that the simplified form is negative one over
two to the 15th power, which is option (E).
In this example, we have two
different variables and negative exponents.
Simplify 𝑚 over 𝑛 to the negative
one power all taken to the negative third power times two times 𝑚 to the negative
two power over 𝑛 to the negative two power all taken to the negative three
That was quite a mouthful. Let’s go ahead and copy down this
expression. The first thing that we see is that
we have a power of a power. Both of these fractions are being
taken to the negative three power. On top of that, we know that when
we’re dividing values and they’re being taken to the same power, we can distribute
that power. Which means for this first
fraction, I can write 𝑚 to the negative three power over 𝑛 to the positive three
This is positive three because
negative one times negative three is positive three. For the second fraction, we have
two to the negative three power and then 𝑚 to the negative two times negative three
power, which will be 𝑚 to the positive six power. And then, in the denominator, 𝑛 to
the negative two times negative three, 𝑛 to the positive six power.
We know when we’re multiplying
fractions, we can multiply the numerators and multiply the denominators. We can multiply 𝑚 to the negative
three power times 𝑚 to the positive six power, which will be 𝑚 to the negative
three plus six power, 𝑚 cubed. And we still have that two to the
negative three power in the numerator. In the denominator, we have 𝑛
cubed times 𝑛 to the sixth power, which will be 𝑛 to the three plus sixth power,
𝑛 to the ninth power.
We nearly have a simplified
form. We can’t simplify 𝑚 cubed any
further, or 𝑛 to the ninth. However, we still have this two to
the negative three power. And that means we need to bring
that value into the denominator, which gives us 𝑚 cubed over two cubed times 𝑛 to
the ninth. And we know that two cubed is
eight, which makes the simplified form of this expression 𝑚 cubed over eight times
𝑛 to the ninth power.
In our final example, we’re going
to consider something we haven’t seen before, a mixed number as a base for an
Which of the following is equal to
one and three-fifths squared times one and three-fifths to the negative three
power? (A) Five-eighths, (B) 25 over 64,
(C) eight-thirds, (D) negative five-eighths, or (E) negative eight-fifths.
We copy down our expression. Before we get started, we need to
make one very important clarification. We have the rule that 𝑥 times 𝑦
squared is equal to 𝑥 squared 𝑦 squared. You might be tempted to write one
squared times three-fifths squared. This is not true. And that’s because the mixed number
one and three-fifths represents one plus three-fifths. And it does not represent one times
three-fifths. And since we can’t do that kind of
distribution, we need a new strategy to simplify.
We’ll need to convert these mixed
numbers into improper fractions. The improper fraction will be one
times five plus three, which is eight, over the original denominator of five. Both of these values are the same
mixed number, so they’re both the eight-fifths as an improper fraction. Once we’re to this point, since
these exponents have the same base, we can add their powers. Two plus negative three is negative
one. And from there, we are able to
distribute that power so that we have eight to the negative one power over five to
the negative one power.
Since we have a negative power in
the numerator and a negative power in the denominator, we can flip them. Five to the first power is
five. Eight to the first power is
eight. The equivalent expression is
five-eighths, which is option (A). The key to solving this one was
recognizing that you needed to change the format of these mixed numbers before you
went about solving.
From here, we’re ready to summarize
the key points. Simplifying expressions allows to
write expressions in the most compact and efficient manner without changing the
value of the expression. And when these expressions have
negative and fractional exponents, we use these rules to simplify the terms.