Algebra

Common Misconceptions (Algebra)

Algebra Common Misconceptions: Level 2 Challenges

         

What is the value of

(8)2? \large \sqrt{ ( \color{#D61F06}-8\color{#333333}) ^ 2 } \hspace{.15cm} ?

ax+5b=2a+bqimpliesx=2 and q=5.\large {\color{#69047E}a}{\color{#D61F06}x} + {\color{#20A900}5}{\color{#69047E}b} = {\color{#D61F06}2}{\color{#69047E}a} + {\color{#69047E}b}{\color{#20A900}q} \\\\\\ \large \text{implies} \\\\\\ \large {\color{#D61F06}x} = {\color{#D61F06}2} \text{ and } {\color{#20A900}q}={\color{#20A900}5}.

True or False?

For all real numbers a,b,x, and q{\color{#69047E}a}, {\color{#69047E}b}, {\color{#D61F06}x}, \text{ and } {\color{#20A900}q}, the statement above must be true.

1=i(1)11=1i(2)11=1i(3)11=1i(4)11=1i(5)11=1i(6)i=1i(7)i2=1(8)1=1(9)\begin{aligned} \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{aligned}

Consider these steps above.

In which step is the (first) mistake committed?

True or False:

Given any two periodic functions, f(x)\color{#D61F06}{f(x)} and g(x)\color{#3D99F6}{g(x)}, the function h(x)=f(x)+g(x)\color{#69047E}{h(x)} = \color{#D61F06}{f(x)} + \color{#3D99F6}{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)f(x) times the period of g(x)g(x)).

What is the value of

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

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