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Chapter 3: Problem 68

Show that \((-37+30 \sqrt{3})^{1 / 3}=-1+2 \sqrt{3}\).

Short Answer

Expert verified
We have shown that cubing the expression \(-1 + 2\sqrt{3}\) results in \(-37 + 30\sqrt{3}\). Therefore, the cube root of \((-37 + 30\sqrt{3})\) is equal to \(-1 + 2\sqrt{3}\).

Step by step solution

01

Cube the provided expression

In this step, we need to calculate the cube of the expression \(-1 + 2\sqrt{3}\). Let \(a=-1\) and \(b=2\sqrt{3}\). \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
02

Substitute the values and simplify

Now substituting the values of a and b in the above formula, we can calculate the cube of the provided expression: \((a+b)^3 = (-1)^3 + 3(-1)^2(2\sqrt{3}) + 3(-1)(2\sqrt{3})^2 + (2\sqrt{3})^3\) Calculate each term separately: \((-1)^3 = -1\) \(3(-1)^2(2\sqrt{3}) = 6\sqrt{3}\) \(3(-1)(2\sqrt{3})^2 = -36\) \((2\sqrt{3})^3 = 24\sqrt{3}\)
03

Add the calculated terms

Now, let's add all the terms we have calculated in the previous step: \((-1) + 6\sqrt{3} - 36 + 24\sqrt{3} = -37 + 30\sqrt{3}\)
04

Compare the result with the given complex number

As we can see, the cube of the provided expression \((-1 + 2\sqrt{3})^3\) is equal to the given complex number \(-37 + 30\sqrt{3}\). Therefore, we can confidently say the following: \((-37+30\sqrt{3})^{1/3} = -1 + 2\sqrt{3}\) This completes the exercise, and we have shown that the cube root of \((-37 + 30\sqrt{3})\) is equal to \(-1 + 2\sqrt{3}\).

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