Video: Factorizing Expressions Involving Exponents

Which of the following numbers is the expression 3^(π₯ + 5) + 3^(π₯) divisible by for all natural numbers π₯? [A] 493 [B] 489 [C] 245 [D] 244

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Video Transcript

Which of the following numbers is the expression three to the π₯ plus five power plus three to the π₯ power divisible by for all natural numbers of π₯?

Letβs start by looking in our expression. Three to the π₯ plus five power plus three to the π₯ power. If we remember that π₯ to the π power times π₯ to the π power equals π₯ to the π plus π power, we can rewrite our first term. So that it says three to the π₯ power times three to the fifth power. Three to the π₯ plus five power is the same thing as three to the π₯ power times three to the fifth power.

Now we have three to the π₯ power times three to the fifth power plus three to the π₯ power. Notice that we have two terms of three to the π₯ power. We could imagine that our second term was three to the π₯ power times one. Because three to the π₯ power times one is itself. We can undistribute this three to the π₯ power, pull it out. And then we would have three to the π₯ power times three to the fifth power plus one. We havenβt changed the value. These are exactly the same just written in a different format. And from here, we could calculate three to the fifth power, which equals 243. Bring everything else down. 243 plus one equals 244. Weβve rewritten three to the π₯ plus five power plus three to the π₯ power to say three to the π₯ power times 244.

We can use this information to find out which of the following numbers this expression is divisible by. Taking this expression. If we took that value and divided it by 244, answer choice D, we would get three to the π₯ power. And that means for any natural number π₯, three to the π₯ plus five power plus three to the π₯ power is divisible by 244. Another way to put this is that 244 is a multiple of this expression. And if 244 is a multiple of this expression, then that expression will always be divisible by 244. And option D, 244, is correct.