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Chapter 3: Problem 57

Find an integer \(m\) such that $$ \left((3+2 \sqrt{5})^{2}-m\right)^{2} $$ is an integer.

Short Answer

Expert verified
The integer \(m\) that makes the entire expression an integer is \(m = 12\sqrt{5}\).

Step by step solution

01

Simplify \((3+2\sqrt{5})^2\)

In order to simplify this expression, we will expand it using the formula \((a+b)^2 = a^2 + 2ab + b^2\). In our case, \(a = 3\) and \(b = 2\sqrt{5}\), so: \((3+2\sqrt{5})^2 = 3^2 + 2(3)(2\sqrt{5}) + (2\sqrt{5})^2\) #Step 2: Calculate the square of each term#
02

Calculate \(3^2\), \((2\sqrt{5})^2\), and \(2(3)(2\sqrt{5})\)

We can calculate the square of each term separately as follows: \(3^2 = 9\) \((2\sqrt{5})^2 = (2)^2(\sqrt{5})^2 = 4\cdot 5 = 20\) \(2(3)(2\sqrt{5}) = 12\sqrt{5}\) #Step 3: Combine the terms#
03

Combine the terms from Step 2

Using the results from Step 2, we can now combine the terms: \((3+2\sqrt{5})^2 = 9 + 20 + 12\sqrt{5} = 29 + 12\sqrt{5}\) #Step 4: Subtract m and obtain an integer#
04

Determine \(m\)

We want to find an integer \(m\) such that: \(({(29 + 12\sqrt{5}) - m})^2\) is an integer. Since we already have one integer term, 29, our focus should be on making the term with the square root, \(12\sqrt{5}\), an integer as well. To do this, we can set \(m\) equal to this term, such that: \(m = 12\sqrt{5}\) Now, substituting the value of \(m\) in the expression: \(({(29 + 12\sqrt{5}) - (12\sqrt{5})})^2 = (29)^2 = 841\) Since 841 is an integer, our expression becomes: \(\left((29)^2\right)^2 = 841^2\) #Step 5: Check the result#
05

Verify that the result is an integer

Now that we have found the value for \(m\) that makes the entire expression an integer, let's check to see if it is indeed an integer: \(\left((3+2\sqrt{5})^{2}-12\sqrt{5}\right)^{2} = \left((29 + 12\sqrt{5}) - 12\sqrt{5}\right)^2 = (29)^2 = 841\) Since 841 is an integer, we have successfully found the value for \(m\): \(m = 12\sqrt{5}\).

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