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# QOTD #1 81^3 + 27^4 is equivalent to which

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QOTD #1 81^3 + 27^4 is equivalent to which [#permalink]  27 Jun 2016, 08:42
Expert's post
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Question Stats:

42% (01:30) correct 57% (01:14) wrong based on 19 sessions
$$81^3$$ + $$27^4$$ is equivalent to which of the following expressions?

Indicate all such expressions.

I) ($$3^7$$)($$2$$)

II) ($$3^{12}$$)($$2$$)

III) ($$9^6$$)($$2$$)

IV) $$9^{12}$$

V) $$3^{24}$$

[Reveal] Spoiler: OA
II and III
[Reveal] Spoiler: OA

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Re: QOTD #1 81^3 + 27^4 is equivalent to which [#permalink]  04 Jul 2016, 21:20
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The original expression can be written as 3^24; Choice ii 3^12(2) = 3^24 and Choice iii 9^6(2) = 3^12(2) = 3^24 are equivalent to original expression,hence answers
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Re: QOTD #1 81^3 + 27^4 is equivalent to which [#permalink]  05 Apr 2017, 12:07
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Expert's post
Carcass wrote:
$$81^3$$ + $$27^4$$ is equivalent to which of the following expressions?

Indicate all such expressions.

I) ($$3^7$$)($$2$$)

II) ($$3^{12}$$)($$2$$)

III) ($$9^6$$)($$2$$)

IV) $$9^{12}$$

V) $$3^{24}$$

[Reveal] Spoiler: OA
II and III

Scanning the answer choices, I can see that I should try to rewrite the original expression with base 3 and base 9.

BASE 3

$$81^3$$ + $$27^4$$ = $$(3^4)^3$$ + $$(3^3)^4$$

= $$3^{12}$$ + $$3^{12}$$

= ($$2$$)($$3^{12}$$)

= II

BASE 9
Take the result from above:

$$81^3$$ + $$27^4$$ = ($$2$$)($$3^{12}$$)

Notice that $$3^{12}$$ can be rewritten as $$(3^2)^6$$, and also notice that $$3^2 = 9$$

So, we get: ($$2$$)($$3^{12}$$) = $$(2)$$$$(3^2)^6$$

= ($$2$$)($$9^6$$)

= III

[Reveal] Spoiler:
II and III

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Re: QOTD #1 81^3 + 27^4 is equivalent to which   [#permalink] 05 Apr 2017, 12:07
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