Video Transcript
In this video, we will learn how to
find cube roots of rational numbers.
We will begin by recalling what we
know about the cube root of a perfect cube. The cube root of a perfect cube 𝑛,
written as shown, is the integer 𝑎 such that 𝑎 cubed is equal to 𝑛. In particular, we have the cube
root of 𝑛 is equal to the cubed root of 𝑎 cubed, which is equal to 𝑎.
We will now consider a useful
property of the cube root of a number by considering the product of two perfect
cubes. This states that if 𝑎 and 𝑏 are
integers, then the cube root of 𝑎 cubed 𝑏 cubed is equal to 𝑎𝑏. We can prove this as follows. When we cube 𝑎 multiplied by 𝑏,
we get 𝑎 cubed 𝑏 cubed. And this means that 𝑎𝑏 is the
cube root of 𝑎 cubed 𝑏 cubed. We can use a similar idea to extend
this definition of the cube root to take the cube roots of the quotients of
integers. If 𝑎 and 𝑏 are integers and 𝑏 is
not equal to zero, then the cube root of 𝑎 cubed over 𝑏 cubed is equal to the cube
root of 𝑎 cubed over the cube root of 𝑏 cubed, which is equal to 𝑎 over 𝑏. We will now look at an example
where we need to use this property to determine the cube root of a rational
number.
Evaluate the cube root of 64 over
343.
In this question, we are trying to
find the cube root of a rational number. We recall that if 𝑎 and 𝑏 are
integers and 𝑏 is nonzero, then the cube root of 𝑎 over 𝑏 is equal to the cube
root of 𝑎 over the cube root of 𝑏. In this question, we have a special
type of this rule since both the numerator and denominator of our fraction are
perfect cubes. In this case, the cube root of 𝑎
cubed over 𝑏 cubed is equal to the cube root of 𝑎 cubed over the cube root of 𝑏
cubed. And this is equal to 𝑎 over
𝑏. We need to calculate the cube root
of 64 and the cube root of 343. We know that four cubed is equal to
64. This means that the cube root of 64
can be rewritten as the cube root of four cubed. And this is equal to four. Seven cubed is equal to 343. And this means that the cube root
of 343 is seven. We can therefore conclude that the
cube root of 64 over 343 is four-sevenths.
In our next example, we will
determine the cube root of a rational number given in decimal form.
Find the value of the cube root of
0.027.
In order to answer this question,
we will consider two methods. In the first one, we will begin by
converting the decimal into a fraction. Using our knowledge of place value,
we know that 0.027 is the same as twenty-seven thousandths. This means that we are trying to
calculate the cube root of twenty-seven thousandths. Next, we recall that if 𝑎 and 𝑏
are integers and 𝑏 is nonzero, then the cube root of 𝑎 over 𝑏 is equal to the
cube root of 𝑎 over the cube root of 𝑏. We can therefore calculate the cube
root of 27 and the cube root of 1000 separately. Since three cubed is equal to 27,
the cube root of 27 is three. Likewise, as 10 cubed is 1000, the
cube root of 1000 is 10. The cube root of 27 divided by the
cube root of 1000 is therefore equal to three over 10 or three-tenths. And this is the value of the cube
root of 0.027. Writing this answer in decimal
form, we have 0.3.
We will now consider a second
method we could use to solve this problem. We begin by writing 0.027 as 27
multiplied by 0.001. As already mentioned, 27 is equal
to three cubed. In the same way, 0.001 is equal to
0.1 cubed. We can therefore rewrite the
original expression as shown. Using the fact that the cube root
of 𝑎 cubed 𝑏 cubed is equal to 𝑎𝑏, we can rewrite the right-hand side of our
equation as three multiplied by 0.1, which once again gives us a final answer of
0.3.
In our next example, we will
determine the length of the sides of a cube from its volume.
Find the side length of a cube
given that its volume is 27 over eight cubic centimeters.
We begin by recalling that a cube
of side length 𝑙 centimeters will have a volume of 𝑙 cubed cubic centimeters. This means that in this question,
𝑙 cubed is equal to 27 over eight or twenty-seven eighths. We note that both 27 and eight are
perfect cubes, since three cubed is equal to 27 and two cubed is equal to eight. We can therefore rewrite our
equation as 𝑙 cubed is equal to three cubed over two cubed. In order to solve this, we take the
cube root of both sides. Recalling that if 𝑎 and 𝑏 are
integers and 𝑏 is nonzero, then the cube root of 𝑎 cubed over 𝑏 cubed is equal to
𝑎 over 𝑏. Then, the right-hand side of our
equation simplifies to three over two. And we can therefore conclude that
the side length of a cube with a volume of 27 over eight cubic centimeters is three
over two centimeters. It is also worth noting we could
write this in decimal form of 1.5 centimeters.
In the final example in this video,
we will determine the radius of a sphere from its volume.
Assuming that the value of 𝜋 is 22
over seven, find the radius of a sphere given its volume is 179.6 recurring cubic
centimeters.
We begin by recalling the formula
for the volume of a sphere. It is equal to four-thirds 𝜋𝑟
cubed. In this question, we are told that
the volume is equal to 179.6 recurring cubic centimeters. We know that 0.6 recurring is equal
to the fraction two-thirds. This means that the volume of the
sphere can be rewritten as 179 and two-thirds cubic centimeters. We can convert this mixed number
into an improper or top-heavy fraction. We do this by multiplying the whole
number by the denominator and then adding the numerator. 179 multiplied by three plus two is
equal to 539. The volume of the sphere is
therefore equal to 539 over three cubic centimeters. Substituting this value into our
formula together with the value of 𝜋 of 22 over seven, we have 539 over three is
equal to four-thirds multiplied by 22 over seven multiplied by 𝑟 cubed. Simplifying the right-hand side
gives us 88 over 21𝑟 cubed.
Next, we can multiply through by
three such that 539 is equal to 88 over seven 𝑟 cubed. Dividing through by 88 over seven
and then dividing the numerator and denominator by 11 gives us 𝑟 cubed is equal to
343 over eight. We can then take the cube root of
both sides of this equation. Noting that 343 and eight are both
perfect cubes, as two cubed is equal to eight and seven cubed is equal to 343, we
have 𝑟 is equal to the cube root of seven cubed over two cubed. Recalling one of the properties of
cube roots that if 𝑎 and 𝑏 are integers and 𝑏 is nonzero, the cube root of 𝑎
cubed over 𝑏 cubed is equal to 𝑎 over 𝑏, this means that 𝑟 is equal to seven
over two, which in decimal form is 3.5. If the volume of a sphere is 179.6
recurring cubic centimeters, then its radius is 3.5 centimeters.
We will now finish this video by
recapping the key points. We saw in this video that if 𝑎 and
𝑏 are integers, then the cube root of 𝑎 cubed 𝑏 cubed is equal to 𝑎𝑏. In the same way, if 𝑎 and 𝑏 are
integers and 𝑏 is not equal to zero, then the cube root of 𝑎 cubed over 𝑏 cubed
is equal to 𝑎 over 𝑏. We also saw that in some questions,
we may need to factor the numerator and denominator in order to determine their cube
roots. In the last two examples, we used
cube roots for geometric applications in order to calculate the side length of a
cube and the radius of a sphere.
