Video: Estimating Values of Irrational Numbers

Given that 𝑥 = √(80) + 78 and 𝑦 = ∛(215) + 37, which of the following is a good estimate of the value of 𝑥 + 𝑦? [A] 130 [B] 129 [C] 56 [D] 131 [E] 44

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Video Transcript

Given that 𝑥 equals root 80 plus 78 and 𝑦 equals cube root of 215 plus 37, which of the following is a good estimate of the value of 𝑥 plus 𝑦?

So in order to solve this problem, what we need to consider are the values of root 80 and the cube root of 215. But before we even need to do that, we can rule out two of our answers straightaway. And those two answers are answer C and answer E, which are 56 and 44. And that’s because if we have 𝑥 plus 𝑦, we’re gonna have root 80 plus 78 plus the cube root of 215 plus 37. Well, if we disregard our root and cube root, we still got 78 and 37. And if we had these together, we’ll get 115. So therefore, our result must be greater than 56 because we still got root 80 and the cube root of 215 to also include.

So to now find out which answer it is, what we’re gonna do is consider our root 80 and cube root of 215. We’re starting with root 80. Well, root 80 is close to root 81. And we know that the root of 81 is equal to nine. So therefore, we can say that root 80 is approximately nine. So we can use that for our estimate. So now, we can move on to the cube root of 215. Because the cube root of 215 is close to the cube root of 216. And we know that 216 is a cube number. And we know, therefore, that the cube root of 216 is gonna be equal to six. And that’s because six multiplied by six multiplied by six gives us 216. So therefore, we can say that the cube root of 215 is gonna be approximately equal to six. So we can use this value in our estimate.

So now, if we substitute these values back into our original equations, we get 𝑥 is approximately equal to nine plus 78 which gives us an estimate for 𝑥 of 87. And 𝑦 is approximately equal to six plus 37 which gives us an estimate for 𝑦 of 43. So therefore, we can say that 𝑥 plus 𝑦 is approximately equal to 87 plus 43 which leaves us with an estimate of 130. So therefore, we can say the given that 𝑥 equals root 80 plus 78 and 𝑦 equals the cube root 215 plus 37, that answer A, 130, is a good estimate of the value of 𝑥 plus 𝑦.

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