Video Transcript
In this video, we’ll learn how to
simplify algebraic expressions and solve algebraic equations involving 𝑛th roots,
where 𝑛 is a positive integer greater than or equal to two.
The 𝑛th root is a really important
mathematical operation. And it describes the inverse of a
power operation with some exponent 𝑛. We define it formally and say that
the 𝑛th root of a number 𝑥, where 𝑛 is a positive integer, is a number that when
raised to the 𝑛th power gives 𝑥. In other words, if we define that
as 𝑦, 𝑥 is equal to 𝑦 to the 𝑛th power. Then the 𝑛th root is as shown. 𝑦 is equal to the 𝑛th root of
𝑥. And whilst it’s outside the scope
of this lesson to investigate in too much detail, it’s worth noting that the 𝑛th
root of 𝑥 can be equivalently written as 𝑥 to the power of one over 𝑛. And this is useful as it can act as
a tool to help us understand how the rules of exponents that we already know can be
applied to expressions involving roots.
The first property of 𝑛th roots
tells us what happens when we multiply them. It says that when the 𝑛th root of
𝑎 and the 𝑛th root of 𝑏 are well defined and are real numbers, then the 𝑛th root
of 𝑎 times 𝑏 is also defined. And it’s simply the product of the
𝑛th root of 𝑎 and the 𝑛th root of 𝑏. Similarly, the quotient of the 𝑛th
root of 𝑎 over the 𝑛th root of 𝑏 is equal to the 𝑛th root of 𝑎 over 𝑏,
assuming that the 𝑛th root of 𝑏 is not equal to zero.
If 𝑛 is an odd integer, then the
𝑛th root of 𝑎 to the 𝑛th power is equal to the 𝑛th root of 𝑎 to the 𝑛th power,
which is simply 𝑎. Similarly, if 𝑛 is even and 𝑎 is
greater than or equal to zero, the 𝑛th root of 𝑎 to the 𝑛th power is equal to
𝑎. But if 𝑛 is even and 𝑎 is less
than zero, then the 𝑛th root of 𝑎 to the 𝑛th power is undefined over the set of
real numbers. However, if 𝑛 is even and 𝑎 is a
real number, we can calculate the 𝑛th root of 𝑎 to the 𝑛th power. And it’s the modulus or absolute
value of 𝑎.
Now, we might be starting to wonder
what on Earth is happening when 𝑛 is even. We will investigate this in more
detail a little bit later on in this video. For now, let’s just look at a
really simple example that will allow us to use these properties to simplify an
expression involving an 𝑛th root.
Simplify the cube root of 64 times
𝑚 cubed.
In order to be able to simplify an
expression involving an 𝑛th root, where here 𝑛 is equal to three, let’s recall one
of the properties that applies to 𝑛th roots. The first property that we’re
interested in says that, for positive real numbers 𝑎 and 𝑏 and positive integers
𝑛, the 𝑛th root of 𝑎 times the 𝑛th root of 𝑏 is equal to the 𝑛th root of
𝑎𝑏. Now, we’re going to apply this in
reverse so that it will allow us to separate 64 and 𝑚 cubed. So, we split 64𝑚 cubed up into 64
and 𝑚 cubed. And using this property, we can
write the cube root of 64𝑚 cubed as the cube root of 64 times the cube root of 𝑚
cubed.
Next, we know that for odd integers
𝑛 the 𝑛th root of 𝑎 to the 𝑛th power is simply equal to 𝑎. Since 𝑛 is three, which is odd, we
can say that the cube root of 𝑚 cubed is just equal to 𝑚. We might also spot that 64 is equal
to four cubed. So, we can write the cube root of
64 as the cube root of four cubed, which is simply four. Let’s substitute each of these
expressions back into our earlier equation, and that will allow us to simplify our
original expression. The cube root of 64 times the cube
root of 𝑚 cubed is simply four 𝑚. And so, our expression is fully
simplified; it’s four 𝑚.
We’ll now repeat this process, but
this time we’ll consider an even root.
Simplify the square root of 100𝑥
to the 16th power.
When a root is given with the value
of 𝑛 omitted, we assume it’s equal to two, and that’s why we define it as the
square root. So, this means we can simplify the
square root of 100𝑥 to the 16th power by applying the property of 𝑛th roots, where
𝑛 is equal to two. This rule tells us that when the
𝑛th root of 𝑎 and the 𝑛th root of 𝑏 are well defined and real numbers, then the
𝑛th root of 𝑎𝑏 is also defined as the 𝑛th root of 𝑎 times the 𝑛th root of
𝑏. We omit the value of 𝑛 and say
that the square root of 𝑎 times the square root of 𝑏 is the square root of
𝑎𝑏.
We’re going to essentially apply
this property in reverse. And it will allow us to separate
the 100 and 𝑥 to the 16th power. We say that the square root of
100𝑥 to the 16th power is the square root of 100 times the square root of 𝑥 to the
16th power. But of course the square root of
100 we know is equal to 10. So, we can rewrite this further as
10 times the square root of 𝑥 to the 16th power. Now, by reversing the
multiplication rule for exponents, we’re going to write 𝑥 to the 16th power as 𝑥
to the eighth power squared. This allows us to apply another
property. And this says that the 𝑛th root of
𝑥 to the 𝑛th power is equal to 𝑥 when 𝑛 is even. So, this allows us to write the
square root of 𝑥 to the 16th power as the square root of 𝑥 to the eighth power
squared, which is simply 𝑥 to the eighth power.
Finally, let’s substitute our
expression back into our earlier equation. And when we do, we find that the
square root of 100𝑥 to the 16th power is 10𝑥 to the eighth power. Now, care must be taken when
finding powers of roots. In this example, we demonstrated
that the square root of 𝑥 to the eighth power squared is equal to 𝑥 to the eighth
power. And this worked because we had an
even power inside an even root. So, the operation was defined for
all real values of 𝑥.
We would need to take great care if
we were simplifying something like the square root of 𝑥 to the power of 14. We could start in a similar
way. We write 𝑥 to the 14th power as 𝑥
to the seventh power squared. But we need to make that equal to
the absolute value of 𝑥 to the seventh power. Now, this is because 𝑥 to the 14th
inside the square root acts first. And that ensures that the radical
is positive for all values of 𝑥. This means that we then take the
even root of a positive number. And so, the resulting function can
only output positive values. Since 𝑥 to the seventh power is
negative if 𝑥 itself is negative, we have to include the absolute value when
simplifying.
Now, we’ve used properties of roots
so far in this video to express an 𝑛th root as the product of two unique 𝑛th
roots. It’s worth noting that we can
extend this property to express the root as the product of three or more 𝑛th
roots. Once again, if the 𝑛th root of 𝑎,
the 𝑛th root of 𝑏, and the 𝑛th root of 𝑐 are well defined, then the product is
equal to the 𝑛th root of 𝑎𝑏𝑐, where 𝑛 is a positive integer. Let’s demonstrate this in our next
example.
Write the square root of 25𝑎
squared 𝑏 to the sixth power in its simplest form.
Remember, if the 𝑛th root of 𝑎,
the 𝑛th root of 𝑏, and the 𝑛th root of 𝑐 are well defined for positive integers
𝑛, their net product is the 𝑛th root of 𝑎𝑏𝑐. Let’s apply this property in
reverse and separate 25𝑎 squared and 𝑏 to the sixth power. We write the square root of 25𝑎
squared 𝑏 to the sixth power as the square root of 25 times the square root of 𝑎
squared times the square root of 𝑏 to the sixth power. Now, of course, the square root of
25 is five. So, we can rewrite this further as
five times the square root of 𝑎 squared times the square root of 𝑏 to the sixth
power.
Next, we’ll use the fact that for
even values of 𝑛 the 𝑛th root of 𝑥 to the 𝑛th power is equal to the absolute
value of 𝑥. This allows us to rewrite the
square root of 𝑎 squared as the absolute value of 𝑎. In a similar way, if we rewrite 𝑏
to the sixth power as 𝑏 cubed squared, this means that the square root of 𝑏 to the
sixth power is the absolute value of 𝑏 cubed. And so, we’re simplifying our root
and we get five times the absolute value of 𝑎 times the absolute value of 𝑏
cubed. But of course, five is
positive. So, the absolute value of five is,
in fact, five. And we know the product of absolute
value is the absolute value of the products. And so, the square root of 25𝑎
squared 𝑏 to the sixth power is the absolute value of five 𝑎𝑏 cubed.
So, we’ve demonstrated how to use
some of the properties of roots to simplify expressions. We need to be a little bit careful
when working with equations that involve exponents. So, consider the equation 𝑦
squared is equal to 16. A solution to this equation is
found by taking the square root of both sides. So, 𝑦 is equal to the square root
of 16, which is equal to four. However, if we substitute 𝑦 equals
negative four into the expression 𝑦 squared, we get negative four squared, negative
four times negative four, which is 16. So, in fact, there’s a second
solution to the equation 𝑦 squared equals 16; it’s negative four.
And so, when solving an equation of
the form 𝑦 squared equals 𝑥, where 𝑥 is a positive real number, the solutions
include both the positive and negative square roots of 𝑥. And so, there’s a real subtle
difference between the two seemingly equivalent statements 𝑦 squared equals 𝑥 and
𝑦 is equal to the square root of 𝑥. In fact, there’s actually a
difference between the two statements 𝑦 to the 𝑛th power equals 𝑥 and 𝑦 equals
the 𝑛th root of 𝑥. Let’s generalize this idea.
Consider the equation 𝑦 to 𝑛th
power equals 𝑥, where 𝑥 and 𝑦 are real numbers and 𝑛 is a positive integer. If 𝑛 is odd, then it doesn’t
matter if 𝑥 is less than or greater than zero. There is always one solution: 𝑦 is
equal to the 𝑛th root of 𝑥. If 𝑛 is even and 𝑥 is negative,
there are, in fact, no real solutions to the equation 𝑦 to the 𝑛th power equals
𝑥. But if 𝑥 is positive, then there
are two solutions to the equation. They are 𝑦 equals the positive and
negative 𝑛th root of 𝑥.
Now, because we interpret the
statements 𝑦 to the 𝑛th power equals 𝑥 and 𝑦 is equal to the 𝑛th root of 𝑥
differently, we have another definition and that’s of the principal 𝑛th root. With this definition, we’re able to
consider an 𝑛th root as a function by making it by definition a one-to-one
mapping. We say that every positive real
number has a single positive 𝑛th root to find the 𝑛th root of 𝑥. This is known as the principal 𝑛th
root. So, when given the expression the
square root of four, we know we’re only interested in two and not negative two
because we’re not essentially solving an equation. Let’s demonstrate an application of
the properties of even and odd 𝑛th roots in our next example.
Find the value, or values, of 𝑥 if
12𝑥 to the fifth power equals 384.
To solve an equation, we apply a
series of inverse operations. We’ll begin by dividing both sides
of this equation by 12. 12𝑥 to the fifth power divided by
12 is just simply 𝑥 to the fifth power. And 384 divided by 12 is equal to
32. So, our equation becomes 𝑥 to the
fifth power is equal to 32. And we recall that for some
equation 𝑥 to the 𝑛th power equals 𝑦, where 𝑥 and 𝑦 are real numbers and 𝑛 is
a positive integer, if 𝑛 is odd, there is exactly one solution. It’s 𝑥 is equal to the 𝑛th root
of 𝑦.
Well, in our equation, 𝑛 is equal
to five. So, it is odd, and there’s one
solution to the equation. It’s found by taking the fifth root
of 32, which is simply equal to two. So, the solution to the equation
12𝑥 to the fifth power equals 384 is 𝑥 equals two.
Let’s consider one more
example. This time, we’re going to combine
the properties of 𝑛th roots and the theorem we just saw. This will allow us to solve more
complicated equations involving exponents.
Find the value, or values, of 𝑥
given that 𝑥 plus nine over five squared equals the square root of 144 times three
squared.
Let’s begin by evaluating the
right-hand side of this equation. We can see that 144 is 12 squared,
and obviously three squared is a square number too. So, we can use a property of roots
to find the principal square root of the product of these two numbers. It’s the square root of 144 times
the square root of three squared. That’s 12 times three, which is
equal to 36. Next, we recall that given some
equation 𝑦 to the 𝑛th power equals 𝑥, where 𝑥 and 𝑦 are real numbers and 𝑛 is
a positive even integer, if 𝑥 is greater than zero, there are two solutions to the
equation. They are 𝑦 equals positive or
negative 𝑛th root of 𝑥.
So, if we rewrite our equation as
𝑥 plus nine over five squared equals 36, then since our 36 is greater than zero and
𝑛 is even — it’s two — we need to take the positive and negative square root of
36. So, 𝑥 plus nine over five is equal
to positive or negative six. We’ll separate this into two
equations. And since we’re going to be
performing the same series of inverse operations, we’ll do them side by side.
We begin by multiplying both sides
of the equation by five. So, our first equation becomes 𝑥
plus nine equals 30, and our second is 𝑥 plus nine equals negative 30. Then, we’ll subtract nine from both
sides on each equation, which gives us 𝑥 equals 21 and 𝑥 equals negative 39. And in fact, we could substitute
either value back into the expression 𝑥 plus nine over five all squared. And it would indeed give us 36 as
we expected. So, the two solutions are 𝑥 equals
21 and 𝑥 equals negative 39.
Let’s finish by recapping some key
points from this lesson. In this lesson, we learned that the
𝑛th root of some number 𝑥, where 𝑛 is a positive integer, is a number that when
raised to the 𝑛th power gives 𝑥. Defining that number as 𝑦, we
write 𝑥 is equal to 𝑦 to the 𝑛th power. Then, the 𝑛th root is written as
shown: 𝑦 is equal to the 𝑛th root of 𝑥. We saw that every positive real
number has a single positive 𝑛th root to find the 𝑛th root of 𝑥. And we called this the principal
𝑛th root. We saw that when the 𝑛th root of
𝑎 and the 𝑛th root of 𝑏 are well defined and real numbers, the 𝑛th root of 𝑎𝑏
is also defined as their product. And if the 𝑛th root of 𝑏 is not
equal to zero, then the same can be said for their quotient. The 𝑛th root of 𝑎 over the 𝑛th
root of 𝑏 is the 𝑛th root of 𝑎 over 𝑏.
If 𝑛 is an odd integer, we saw
that the 𝑛th root of 𝑎 all to the 𝑛th power is equal to the 𝑛th root of 𝑎 to
the 𝑛th power, which is 𝑎. But if 𝑛 is even and 𝑎 is less
than zero, then the 𝑛th root of 𝑎 to the 𝑛th power is undefined over the set of
real numbers. If 𝑎 is greater than zero, it’s
simply equal to 𝑎. But if 𝑛 is even and 𝑎 is any
real number, then the 𝑛th root of 𝑎 to the 𝑛th power is the absolute value of
𝑎. Finally, we saw that solutions to
the equation 𝑦 to the 𝑛th power equals 𝑥, where 𝑥 and 𝑦 are real numbers and 𝑛
is a positive integer, are 𝑦 is equal to the 𝑛th root of 𝑥 if 𝑛 is odd. And if 𝑛 is even but 𝑥 is
negative, there are no real solution. And if it’s positive, the solutions
are the positive and negative 𝑛th root of 𝑥.
