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Bunuel
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4^(6 + 12 + 18 + 24 + … + 6x) = (0.0625)^(-84), what is the value of x 21 Apr 2020, 09:06
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64% (02:56) correct 36% (02:51) wrong based on 282 sessions

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If \(4^{(6 + 12 + 18 + 24 + … + 6x)} = (0.0625)^{(-84)}\), what is the value of x?

A. 6
B. 7
C. 8
D. 9
E. 12

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4^(6 + 12 + 18 + 24 + … + 6x) = (0.0625)^(-84), what is the value of x 21 Apr 2020, 09:18
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Bunuel
If \(4^{(6 + 12 + 18 + 24 + … + 6x)} = (0.0625)^{(-84)}\), what is the value of x?

A. 6
B. 7
C. 8
D. 9
E. 12

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\(4^{(6 + 12 + 18 + 24 + … + 6x)} = (0.0625)^{(-84)}=(\frac{625}{10000})^{(-84)}=((\frac{5}{10})^4)^{-84}=((\frac{1}{2})^4)^{-84}=(2^4)^{84}=4^{2*84}\)

Equating the powers of 4 ..
\((6+12+...6x)=2*84..........6(1+2+3....+x)=2*84\).
\(1+2+...+x=2*14\).
\(\frac{x(x+1)}{2}=2*14\).
\(x(x+1)=2*2*14=2*2*2*2*7=7*8....x=7\)

B
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4^(6 + 12 + 18 + 24 + … + 6x) = (0.0625)^(-84), what is the value of x 02 Aug 2020, 11:03
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[quote="Bunuel"]If \(4^{(6 + 12 + 18 + 24 + … + 6x)} = (0.0625)^{(-84)}\), what is the value of x?

A. 6
B. 7
C. 8
D. 9
E. 12

4^{(6 + 12 + 18 + 24 + … + 6x)} = (0.0625)^{(-84)
4^{(6 + 12 + 18 + 24 + … + 6x)} = (10000/625)^84=4^168
6 + 12 + 18 + 24 + … + 6x=168
6(1+2+3+4....+6x)=168
1+2+3....+x=28

x=7
B
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4^(6 + 12 + 18 + 24 + … + 6x) = (0.0625)^(-84), what is the value of x 22 Aug 2020, 21:14
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.0625 in Fractional Form = 625 / 10, 000

Rule: for (-)Negative Integer Exponents, take the Reciprocal of the Base and Make the Power (+)Positive

(625 / 10, 000) ^ -84 = (10, 000 / 625) ^ 84

10,000 Divided by 625 = 16

(16) ^ 84 = [ (4^2) ] ^84 = 4^168

on the L.H.S. of the Equation, Each Term that is being Added to the Exponent of 4 is a Multiple of 6

1st Multiple of 6 (6)
2nd Multiple of 6 (12)
3rd Multiple of 6 (18) etc.

Since we need the L.H.S. to equate to = 4^168 ---- we need all the Multiple of 6 to add up to 168.

How may Multiple of 6 are there in 168? 168 / 6 = 28

Thus, we need 28 "Multiples of 6s" to have the L.H.S.'s Exponent = 168

Chart of Multiple of 6 in each No. being added:

6 -- 1
12 --- 2
18 --- 3
24 ---- 4
30 -----5
36 ---- 6
42 ---- 7
_______________
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 Multiple of 6

In order for the Exponent on the L.H.S. to SUM to 168, the last term should be 42 = 6 * (7) = 6 * (x)

X = 7
Answer B
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4^(6 + 12 + 18 + 24 + … + 6x) = (0.0625)^(-84), what is the value of x 29 Apr 2024, 12:15
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