Algebra Through Puzzles
# Introduction

**Here are some situations where some quick shortcuts can save you a lot of time and help you avoid annoying arithmetic.**

Simplify \[\large \frac{ {\color{green}{2}} ^ {\color{blue}{3}} \times {\color{green}{2}} ^ {\color{blue}{7}}}{ {\color{green}{2}} ^ {\color{blue}{6}} \times {\color{green}{2}} ^ {\color{blue}{3}} }. \]

Note: You are not a robot, and you don't need a calculator to solve this!

Which is larger,

\[ \large 3 ^ { 4 } + 3 ^ { 4 } + 3 ^ { 4 } \hspace{.5cm} \text{ or } \hspace{.55cm} {3} ^ { 5} \, ? \]

Evaluate

\[ \large \frac{ \color{blue}{42} ^ 2 - \color{green}{38} ^ 2 } { \color{blue}{42} - \color{green}{38} }. \]

Hint: Use the difference of two squares identity:

At which step is the first error made in solving the equation given in Step 1?

**Step 1.** \(\,\,\,\sqrt{x + 4} = 5\)

**Step 2.** \(\,\,\,\sqrt{x} + \sqrt{4} = 5\)

**Step 3.** \(\,\,\,\sqrt{x} + 2 = 5\)

**Step 4.** \(\,\,\,\sqrt{x} = 3\)

**Step 5.** \(\,\,\,x = 9\)

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