The easiest and fastest solution is by this way: we must only notice that we deal with geometric series with increasing exponents. Then we must find the sum of that geometric series. The first exponent of the first 2 is 2^(1/2). The second exponent of the first 54 is 54^(1/4). So the first pair (2 and 54) we can rewrite like that: 2^(1/2) * 54^(1/4) = 2^(1/4) * 2^(1/4) *54^(1/4) = (2*2*54)^(1/4) = 216^(1/4) the second pair (2 and 54) we can rewrite like that: 2^(1/8) * 54^(1/16) = 2^(1/16) * 2^(1/16) *54^(1/16) = (2*2*54)^(1/16) = 216^(1/16) the third pair (2 and 54) we can rewrite like that: 2^(1/32) * 54^(1/64) = 2^(1/64) * 2^(1/64) *54^(1/64) = (2*2*54)^(1/64) = 216^(1/64) We have 216^(1/4) *216^(1/16) * 216^(1/64)... and so on For that geometric series a = 1/4 and r = 1/4 and n = 216 The formula: Sum = n^ [a / (1-r)] Sum = 216^(1/4) / (1 -( 1/4)) Sum = 216^ (1/4) : (3/4) = 216^ (1/4) * (4/3) = 216^1/3 216^1/3 = 6
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Nice, What an innovative way to reach at the conclusion.
The easiest and fastest solution is by this way:
we must only notice that we deal with geometric series with increasing exponents.
Then we must find the sum of that geometric series.
The first exponent of the first 2 is 2^(1/2).
The second exponent of the first 54 is 54^(1/4).
So the first pair (2 and 54) we can rewrite like that:
2^(1/2) * 54^(1/4) = 2^(1/4) * 2^(1/4) *54^(1/4) = (2*2*54)^(1/4) = 216^(1/4)
the second pair (2 and 54) we can rewrite like that:
2^(1/8) * 54^(1/16) = 2^(1/16) * 2^(1/16) *54^(1/16) = (2*2*54)^(1/16) = 216^(1/16)
the third pair (2 and 54) we can rewrite like that:
2^(1/32) * 54^(1/64) = 2^(1/64) * 2^(1/64) *54^(1/64) = (2*2*54)^(1/64) = 216^(1/64)
We have
216^(1/4) *216^(1/16) * 216^(1/64)... and so on
For that geometric series
a = 1/4 and r = 1/4 and n = 216
The formula:
Sum = n^ [a / (1-r)]
Sum = 216^(1/4) / (1 -( 1/4))
Sum = 216^ (1/4) : (3/4) = 216^ (1/4) * (4/3) = 216^1/3
216^1/3 = 6
How to solve without a calculator
X= 6
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