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Radical Expressions and Equations

Combine your skills of working with exponents and manipulating expressions for radical glory.

Level 1

         

\[ \Large 2^{\color{red}3} + 2^{\color{red}3} + 3^{\color{blue}4} + 3^{\color{blue}4} + 3^{\color{blue}4} = \ \color{teal} ? \]

If half of \(\sqrt{2}\) is \(\sqrt{k}\), then what is the value of \(k?\)

\[\large \sqrt{\frac{1}{\sqrt{ x}}}=3, \ \ \ x = \ ? \]

\[ \large \frac{\sqrt{12}-\sqrt{27}}{6}\times\sqrt{3}\times\sqrt{4}= \ ? \]

\[ \large\color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} + \color{green} 6^{\color{blue} 6} = \color{green}6^ {\color{brown}a} \]

If \(\color{brown} a\) satisfies the equation above, what is the value of \(\color{brown} a\)?

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