Video Transcript
Express the cube root of 256 over
two minus the cube root of four over 27 in its simplest form.
Recall that when taking any root of
a quotient, such as a cube root, we take the cube root of the numerator over the
cube root of the denominator. If the numerator and denominator
are expressed as 𝑎 cubed and 𝑏 cubed, respectively, then the result is 𝑎 over
𝑏.
In this question, for the first
term, we are taking the cube root of the numerator only. For the second term, we are taking
the cube root of the whole quotient, four over 27. So we can rewrite this as the cube
root of four over the cube root of 27. Four is not a perfect cube, so
we’ll leave this expressed as the cube root of four. 27, however, is a perfect cube,
three cubed. So we can rewrite the cube root of
27 as three. 256 is not a perfect cube
either. But to simplify this expression, we
can look for a factor of the cube root of four and see if we can express the
difference of the two quotients as a single quotient. To do this, we can try to find a
factor of four in 256. 256 is equal to four times 64. So we can rewrite the cube root of
256 as the cube root of four times 64.
Now recall that when taking the
root of a product of multiple terms, such as a cube root, we can take the cube root
of each term and multiply them together. If we express both terms as cubes,
𝑎 cubed and 𝑏 cubed, then the result is 𝑎𝑏. Again, four is not a perfect
cube. So we will leave this as the cube
root of four, giving us a factor of cube root four in both terms. 64, however, is a perfect cube,
four cubed. So taking its cube root gives us
four. The expression therefore becomes
the cube root of four times four over two minus the cube root of four over
three. We now have a common factor of the
cube root of four in both terms.
The remaining factor in the first
term is four over two, which is two, and the remaining factor in the second term is
negative one over three. So we have the cube root of four
times two minus one-third. Two is equal to six-thirds, so this
becomes the cube root of four times six-thirds minus one-third. This is equal to five over
three.
So our final answer for the
simplest form of the cube root of 256 over two minus the cube root of four over 27
is five times the cube root of four over three.
