Video Transcript
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.
