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Chapter 2: Problem 99

Describe the error. $$\sqrt{-6} \sqrt{-6}=\sqrt{(-6)(-6)}=\sqrt{36}=6$$

Short Answer

Expert verified
The error in the statement \(\sqrt{-6} \sqrt{-6} = \sqrt{(-6)(-6)} = \sqrt{36} = 6\) is in assuming that the product of the square roots of negative numbers can be represented as square root of the product of those numbers. When dealing with square roots of negative numbers, the imaginary unit 'i' is involved. So the correct solution should be \(\sqrt{-6} \sqrt{-6} = -6\).

Step by step solution

01

Interpreting the expression

First, look at the expression \(\sqrt{-6} \sqrt{-6}\). Here we have the square root of a negative number times the square root of a negative number. The square root of negative numbers is not defined for real numbers and thus needs to be represented as multiplication of real and imaginary numbers, i.e., in the form of 'a * i', where 'i' is an imaginary unit with the property \(i^2 = -1.\)
02

Applying the rules of imaginary numbers

Considering 'i' as the imaginary unit, we can rewrite \(\sqrt{-6}\) as \(\sqrt{6} \cdot i\). Thus \(\sqrt{-6} \sqrt{-6}\) becomes \((\sqrt{6} \cdot i) (\sqrt{6} \cdot i)\). When these are multiplied, we get \((\sqrt{6})^2 \cdot (i)^2 = 6 \cdot -1 = -6\). You can see that the result contradicts with the original exercise result of \(6\).
03

Identifying the error

In the original statement \(\sqrt{-6} \sqrt{-6} = \sqrt{(-6)(-6)} = \sqrt{36} = 6\), the error lay in assuming \(\sqrt{-6} \sqrt{-6}\) could be simplified as \(\sqrt{(-6)(-6)}\). This would only be true for nonnegative real numbers. For negative or imaginary numbers, the rule doesn't hold because the multiplication of square roots of negative numbers involves the imaginary unit 'i'.

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