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# 10^k + 5^k =

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Math Expert
Joined: 02 Sep 2009
Posts: 56238

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22 Mar 2018, 23:46
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Difficulty:

25% (medium)

Question Stats:

76% (00:29) correct 24% (00:35) wrong based on 66 sessions

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$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

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Joined: 26 Feb 2016
Posts: 3360
Location: India
GPA: 3.12
Re: 10^k + 5^k =  [#permalink]

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23 Mar 2018, 00:00
Bunuel wrote:
$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

Formula used: $$(a*b)^m = a^m * b^m$$

$$10^k + 5^k = (5*2)^k + 5^k = 5^k*2^k + 5^k = 5^k(2^k + 1)$$ (Option B)
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Posts: 1074
Re: 10^k + 5^k =  [#permalink]

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23 Mar 2018, 00:04
Bunuel wrote:
$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

We'll show two approaches to solution:

The first relies on simplification and is a Precise approach.
Since 10^k = (5*2)^k = (5^k)(2^k) then we can extract a common factor of 5^k
from the original equation to get (5^k)(2^k+1).

If we don't feel too confident with the algebraic manipulations, we can instead just pick easy numbers.
This is an Alternative approach.

Say k = 2 then 10^k + 5^k = 100 + 25 = 125.
A. 5(4 + 1) = 25 NO
B. 25(4 + 1) = 125 Maybe, let's check the others.
C. 25*9 = 225 NO
D. 15^2 = 225 NO
E. 15^4 definitely not.

Either way, (B) is our answer.
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Joined: 21 May 2013
Posts: 655
Re: 10^k + 5^k =  [#permalink]

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23 Mar 2018, 05:39
Bunuel wrote:
$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

=5^k(2^k+1)
Manager
Joined: 28 Jan 2018
Posts: 51
Location: Netherlands
Concentration: Finance
GMAT 1: 710 Q50 V36
GPA: 3

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23 Mar 2018, 09:54
Bunuel wrote:
$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

Key formula to use: $$(a*b)^{m} = a^{m} * b^{m}$$

$$10^{k} + 5^{k} = 2^{k}*5^{k} + 5^{k} = 5^{k}*(2^{k} + 1) (B)$$
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2823
Re: 10^k + 5^k =  [#permalink]

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26 Mar 2018, 17:35
Bunuel wrote:
$$10^k + 5^k =$$

A. $$5(2^k + 1)$$

B. $$5^k(2^k + 1)$$

C. $$5^k*3^k$$

D. $$15^k$$

E. $$15^{2k}$$

10^k + 5^k = 2^k * 5^k + 5^k = 5^k(2^k + 1)

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Re: 10^k + 5^k =   [#permalink] 26 Mar 2018, 17:35
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