### Video Transcript

Which of the integers below is
closest to the cube root of negative 353? Is it (A) negative 17, (B) negative
six, (C) negative 18, or (D) negative seven?

Let 𝑥 equal the cube root of
negative 353. If we cube both sides of this, we
have 𝑥 cubed equals negative 353. Therefore, when the question is
asking us to find the integer closest to the cube root of negative 353, we are
equivalently trying to find which of these numbers cubed is closest to negative
353.

Now, we could start by cubing each
of these numbers in turn. However, let us be prudent and
leave the larger negative numbers until last, since they will be more difficult to
calculate. So, let us first consider negative
six cubed. By definition, this is negative six
times negative six times negative six. Multiplying the first two negative
sixes gives us 36. Then, we can split up the
subsequent multiplication to make it easier, giving us negative 180 minus 36, which
gives us negative 216. Comparing this to negative 353,
negative six cubed is not negative enough.

So, let us now consider negative
seven cubed, which is negative seven times itself three times. And this is 49 times negative
seven, which, using whatever multiplicative method we please, we can find is
negative 343, which we can once again compare to the given number and find that our
result is still not negative enough, although we are now a lot closer.

Now, we could explicitly calculate
the cube of negative 17 and negative 18 to prove that they are indeed not closer to
negative 353. However, instead, we could make use
of the fact that if a number 𝑎 is less than another number 𝑏, then 𝑎 cubed will
always be less than 𝑏 cubed. Therefore, since both negative 18
and negative 17 are less than negative 10, then negative 18 and negative 17 cubed
will both be less than negative 10 cubed.

Why does this matter? Well, if we calculate negative 10
cubed, which is just negative 10 times negative 10 times negative 10, which is
negative 1000, then since negative 1000 is clearly further away from negative 353
than negative seven cubed, we can conclude that both negative 18 cubed and negative
17 cubed must be even further away, since they are both even more negative than
negative 10 cubed.

To sketch a number line of the
situation, it looks something like this, where the cube root of negative 353 is
somewhere between negative seven and negative 10 but much closer to negative
seven. Thus, the closest integer to the
cube root of negative 353 from this list is option (D), negative seven.