Video Transcript
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.
