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# If 5^2^5 * 4^1^3 = 2 * 10^k, what is the

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If 5^2^5 * 4^1^3 = 2 * 10^k, what is the [#permalink]  29 Aug 2018, 06:20
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If $$5^2^5 * 4^1^3 = 2 * 10^k$$, what is the value of K

src: Orbit Test Prep

[Reveal] Spoiler:
25
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Re: If 5^2^5 * 4^1^3 = 2 * 10^k, what is the [#permalink]  29 Aug 2018, 07:13
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amorphous wrote:
If $$5^2^5 * 4^1^3 = 2 * 10^k$$, what is the value of K

src: Orbit Test Prep

[Reveal] Spoiler:
25

Useful rule: (xy)^n = (x^n)(y^n)

Given: (5^25)(4^13) = (2)(10^k)

Rewrite 4 as 2^2 to get: (5^25)[(2^2)^13] = (2)(10^k)
Apply power of a power rule to get: (5^25)(2^26) = (2)(10^k)
Rewrite 10 as (5)(2) to get: (5^25)(2^26) = (2)[(5)(2)]^k
Apply above rule to get: (5^25)(2^26) = (2)(5^k)(2^k)

At this point, we can focus solely on the 5's and see that it must be true that 5^25 = 5^k, which means k = 25

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Re: If 5^2^5 * 4^1^3 = 2 * 10^k, what is the [#permalink]  31 Aug 2018, 22:07
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Re: If 5^2^5 * 4^1^3 = 2 * 10^k, what is the [#permalink]  02 Sep 2018, 02:19
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5^25 * 4^13 = 2 * 10^k

simplify both sides as follows:
5^25 * 2 ^ 26 = 2 * 2^k * 5^k
=> 5^25 * 2 ^ 26 = 2^(k+1) * 5^k

Now you can equate either the exponent of 5 or 2.

Equating 5's exponent gives k=25
Equating 2's exponent gives k+1=26 => k=25

Hence, k=25
Re: If 5^2^5 * 4^1^3 = 2 * 10^k, what is the   [#permalink] 02 Sep 2018, 02:19
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