Video Transcript
In this video, we will learn how to
use the properties of operations on cube roots to simplify expressions.
We will begin by recalling one of
the key properties we need to use in this video. The product of cube roots states
that for any real numbers 𝑎 and 𝑏, the cube root of 𝑎 multiplied by the cube root
of 𝑏 is equal to the cube root of 𝑎 multiplied by 𝑏. For example, let’s assume we need
to multiply the cube root of 12 by the cube root of 18. Using the rule above, this can be
rewritten as the cube root of 12 multiplied by 18. And since 12 multiplied by 18 is
216, we have the cube root of 216. As six cubed is equal to 216, then
216 is a perfect cube such that the cube root of 216 is six.
We can also simplify a cube root
which is divisible by a perfect cube by splitting it into the product of cube
roots. For example, we can rewrite the
cube root of 54 as the cube root of 27 multiplied by two. This in turn can be written as the
cube root of 27 multiplied by the cube root of two. Finally, as the cube root of 27 is
three, the cube root of 54 written in its simplest form is three multiplied by the
cube root of two.
By using this second method, we can
rewrite the cube root of a nonperfect cube in a simplified form. Given any integer 𝑐, we write the
cube root of 𝑐 in the form 𝑎 multiplied by the cube root of 𝑏, where 𝑏 is the
smallest possible positive integer, by using the following steps. Firstly, we find the largest
perfect cube which divides 𝑐. Secondly, we choose the integer 𝑎
to have the same sign as 𝑐 such that 𝑎 cubed divides 𝑐 and 𝑎 cubed is the
largest perfect cube dividing 𝑐. We then have that the cube root of
𝑐 is equal to the cube root of 𝑎 cubed multiplied by 𝑏, which is equal to 𝑎
multiplied by the cube root of 𝑏.
We will now consider a couple of
examples of this type.
Write each of the following
radical expressions in the form 𝑎 multiplied by the cube root of 𝑏, where 𝑎
and 𝑏 are integers and 𝑏 is the smallest possible positive value: the cube
root of 256 and the cube root of negative 540.
We begin the first part of this
question by looking for perfect cubes that divide 256. The first six perfect cubes are
one, eight, 27, 64, 125, and 216. The largest of these that
divides exactly into 256 is 64. Recalling the product rule for
cube roots, for any two real numbers 𝑚 and 𝑛 as shown, we can rewrite the cube
root of 256 as the cube root of 64 multiplied by the cube root of four. And since the cube root of 64
is four, the cube root of 256 is equal to four multiplied by the cube root of
four. This is written in the correct
form as required.
We can answer the second part
of the question using the same method. This time, however, we note
that we are trying to cube root a negative number. Since 27 is the largest perfect
cube that divides into 540, we can rewrite the cube root of negative 540 as the
cube root of negative 27 multiplied by the cube root of 20. The cube root of negative 27 is
negative three. And as such, our answer
simplifies to negative three multiplied by the cube root of 20.
In our next example, we will apply
the product of cube roots property to simplify a product of radicals.
Express the cube root of four
multiplied by the cube root of negative 16 in its simplest form.
In order to answer this question,
we first recall that for any two real numbers 𝑎 and 𝑏, the cube root of 𝑎
multiplied by the cube root of 𝑏 is equal to the cube root of 𝑎 multiplied by
𝑏. This means that the cube root of
four multiplied by the cube root of negative 16 is equal to the cube root of four
multiplied by negative 16. Four multiplied by negative 16 is
negative 64. So our expression simplifies as
shown. Next, we note that negative four
cubed is equal to negative 64. And as such, the cube root of
negative 64 is negative four.
So far in this video, we have
worked with simplifying the cube roots of integers. However, this result works with the
cube roots of any real numbers. An application of this result is to
consider the cube root of the quotient of real numbers. For any real numbers 𝑎 and 𝑏,
where 𝑏 is nonzero, we have that the cube root of 𝑎 over 𝑏 is equal to the cube
root of 𝑎 divided by the cube root of 𝑏. We will now look at an example of
using both properties to simplify an expression involving radicals.
Simplify the cube root of two
multiplied by the cube root of four divided by the cube root of 32 multiplied by the
cube root of negative two.
We begin by noting that none of the
numbers here are perfect cubes. And as such, we cannot directly
evaluate any of the individual radicals. Instead, we will use two properties
of cube roots. Firstly, the cube root of 𝑎
multiplied by the cube root of 𝑏 is equal to the cube root of 𝑎 multiplied by
𝑏. This holds if 𝑎 and 𝑏 are real
numbers, as in this case. We can rewrite the numerator of our
fraction as the cube root of two multiplied by four. And the denominator simplifies to
the cube root of 32 multiplied by negative two. This in turn gives us the cube root
of eight over the cube root of negative 64.
Next, we use the fact that when 𝑎
and 𝑏 are real numbers and 𝑏 is nonzero, the cube root of 𝑎 over the cube root of
𝑏 is equal to the cube root of 𝑎 divided by 𝑏. We can therefore rewrite our
expression as the cube root of eight over negative 64. The fraction eight over 64 written
in its simplest form is one-eighth. This means that our expression
simplifies to the cube root of negative one-eighth.
Finally, since negative one-half
cubed is negative one-eighth, then the cube root of negative one-eighth is negative
one-half. The expression the cube root of two
multiplied by the cube root of four divided by the cube root of 32 multiplied by the
cube root of negative two in its simplest form is negative one-half.
In our next example, we will
simplify an expression involving the addition and subtraction of multiple radical
expressions.
Express negative five multiplied by
the cube root of 192 plus five multiplied by the cube root of negative 648 plus the
cube root of 375 in its simplest form.
We will begin this question by
simplifying each of the three terms separately. We will do this by using the fact
that for any real numbers 𝑎 and 𝑏, the cube root of 𝑎 multiplied by the cube root
of 𝑏 is equal to the cube root of 𝑎 multiplied by 𝑏. The largest perfect cube divisor of
192 is 64. Therefore, the cube root of 192 is
equal to the cube root of 64 multiplied by three. This can be written as the cube
root of 64 multiplied by the cube root of three, which in turn is equal to four
multiplied by the cube root of three. The first term in our initial
expression is therefore equal to negative five multiplied by four multiplied by the
cube root of three.
We can use the same process to
simplify the cube root of negative 648. This time, since negative 216
multiplied by three is equal to negative 648, we have the cube root of negative 216
multiplied by three. Using the product of cube roots
property and our knowledge of perfect cubes, we have negative six multiplied by the
cube root of three. The second term in our initial
expression is therefore equal to five multiplied by this.
Next, we can rewrite the cube root
of 375 as the cube root of 125 multiplied by the cube root of three. And this is equal to five
multiplied by the cube root of three. We are now in a position where we
can simplify each of the first two terms. The entire expression simplifies to
negative 20 multiplied by the cube root of three minus 30 multiplied by the cube
root of three plus five multiplied by the cube root of three. Finally, since negative 20 minus 30
plus five is equal to negative 45, we can conclude that the initial expression
written in its simplest form is negative 45 multiplied by the cube root of
three.
We will now finish this video by
recapping the key points. We saw in this video that for any
two real numbers 𝑎 and 𝑏, the cube root of 𝑎 multiplied by the cube root of 𝑏 is
equal to the cube root of 𝑎 multiplied by 𝑏. In the same way, for any two real
numbers 𝑎 and 𝑏, where 𝑏 is nonzero, the cube root of 𝑎 over 𝑏 is equal to the
cube root of 𝑎 divided by the cube root of 𝑏. Finally, we saw that if 𝑐 is an
integer, we can use these results to write the cube root of 𝑐 as 𝑎 multiplied by
the cube root of 𝑏, where 𝑏 is the smallest positive integer and has no perfect
cube divisors greater than one. This is called the simplest form of
the cube root of 𝑐.
