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Algebra Through Puzzles

# Exponent Shortcuts and Misconceptions

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:

What is the value of $\LARGE {\color{blue}{ 2}} ^ { {\color{green}{8}} ^{ {\color{blue} { 2} }} } ?$

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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