Question Video: Finding the Cube Root of Integers Using Estimation | Nagwa Question Video: Finding the Cube Root of Integers Using Estimation | Nagwa

# Question Video: Finding the Cube Root of Integers Using Estimation Mathematics • Second Year of Preparatory School

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Which of the integers below is closest to ∛−353? [A] −17 [B] −6 [C] −18 [D] −7

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### 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.

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