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Algebra

# Algebra Common Misconceptions: Level 2 Challenges

What is the value of

$\large \sqrt{ ( \color{red} {- 8} ) ^ 2 } \hspace{.15cm} ?$

$\large {\color{purple}a}{\color{red}x} + {\color{green}5}{\color{purple}b} = {\color{red}2}{\color{purple}a} + {\color{purple}b}{\color{green}q} \\\\\\ \large \text{implies} \\\\\\ \large {\color{red}x} = {\color{red}2} \text{ and } {\color{green}q}={\color{green}5}.$

True or false?

For all real numbers $${\color{purple}a}, {\color{purple}b}, {\color{red}x}, \text{ and } {\color{green}q}$$, the statement above must be true.

$$$\begin{split} \sqrt{-1}=&i \quad&\ldots(1) \\ \dfrac{1}{\sqrt{-1}}=&\dfrac{1}{i}\quad& \ldots(2) \\ \dfrac{\sqrt {1}}{\sqrt{-1}}=&\dfrac{1}{i} \quad &\ldots(3) \\ \sqrt{\dfrac{1}{-1}}=&\dfrac{1}{i} \quad &\ldots(4) \\ \sqrt{\dfrac{-1}{1}}=&\dfrac{1}{i}\quad &\ldots(5) \\ \dfrac{\sqrt{-1}}{1}=&\dfrac{1}{i} \quad &\ldots(6) \\ i=&\dfrac{1}{i} \quad& \ldots(7) \\ i^2=&1 \quad & \ldots (8) \\ -1=&1\quad & \ldots(9) \end{split}$$$

Consider these steps above.

In which step is the (first) mistake committed?

True or False:

Given any two periodic functions, $$\color{red}{f(x)}$$ and $$\color{blue}{g(x)}$$, the function $$\color{purple}{h(x)} = \color{red}{f(x)} + \color{blue}{g(x)}$$ will also be periodic with a period that is, at most, the product of the original two periods (the period of $$f(x)$$ times the period of $$g(x)$$).

What is the value of

$\large \sqrt{1\text{%}}\, ?$

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