ShmoopTube

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And here's your shmoop du jour.

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Which of the following is equivalent to 2 to the 9th over 16 squared?  

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And here are your potential answers...

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OK this question is really testing whether we can manipulate exponential multiplication.

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There is a brute way of doing this problem -- we could manually multiply two 9 times

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and then square 16...

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...but, as they say in Silicon Valley, "that won't scale."

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Meaning.... What if they gave us 2 to the 105th over 16 squared?

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It'd take us all night to manually do the problem.

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And we need our beauty rest.

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Instead, we need to learn how to set things up so that we can just add and subtract the exponents.

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Life is easier with short cuts. Just ask Britney Spears.

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The key thing we're supposed to be able to recognize in this problem is that we need

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2 as the base instead of 16.

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If we have 2 as the base, then we can just add and subtract the exponents with little

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hassle from The Man.

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So... 2 to the 2nd is 4... 2 to the 3rd is 8, and... voila. 2 to the 4th is 16.

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We can rewrite 16 as 2 to the 4th.

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Now the problem looks like this...

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...and hopefully a little birdie just told you that when dividing exponents we can simply

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subtract terms...

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...and when taking any complex number -- i.e., one that is an exponent -- to a higher power...

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...we just multiply the terms.

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Here, the 2 to the 4th squared becomes 2 to the 2-times-4... or 2 to the 8th.

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And because we are dividing 2 to the 9th by 2 to the 8th, we can just subtract 8 from

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9 and get 2 to the first or... 2.

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Done. Answer: A.

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It's always nice to finish the day with a little number 2.

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