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Math Olympiad Questions: {(³√100 - 1)/[3(³√10) + 3]}⁵ =?Let: a = ³√10; (³√100 - 1)/[3(³√10) + 3] = (a² - 1)/(3a + 3) = (a - 1)/3{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(a - 1)/3]⁵ = [(a - 1)⁵]/(3⁵)(a - 1)⁵ = a⁵ - 5a⁴ + 10a³ - 10a² + 5a - 1 = ³√10⁵ - 5(³√10)⁴ + 10(³√10)³ - 10(³√10)² + 5(³√10) - 1 = 10(³√10²) - 50(³√10) + 100 - 10(³√10²) + 5(³√10) - 1 = 99 - 45(³√10) {(³√100 - 1)/[3(³√10) + 3]}⁵ = [99 - 45(³√10)]/3⁵ = [11 - 5(³√10)]/27 Answer check:{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(a - 1)/3]⁵ = [(√10 - 1)/3]⁵ = (1.154/3)⁵ = 0.385⁵ = 0.0084[11 - 5(³√10)]/27 = [11 - 5(2.154)]/27 = 0.228/27 = 0.084; ConfirmedThe calculation was achieved on a smartphone with a standard calculator appFinal answer:{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(³√10 - 1)/3]⁵ = [11 - 5(³√10)]/27
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Math Olympiad Questions: {(³√100 - 1)/[3(³√10) + 3]}⁵ =?
Let: a = ³√10; (³√100 - 1)/[3(³√10) + 3] = (a² - 1)/(3a + 3) = (a - 1)/3
{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(a - 1)/3]⁵ = [(a - 1)⁵]/(3⁵)
(a - 1)⁵ = a⁵ - 5a⁴ + 10a³ - 10a² + 5a - 1
= ³√10⁵ - 5(³√10)⁴ + 10(³√10)³ - 10(³√10)² + 5(³√10) - 1
= 10(³√10²) - 50(³√10) + 100 - 10(³√10²) + 5(³√10) - 1 = 99 - 45(³√10)
{(³√100 - 1)/[3(³√10) + 3]}⁵ = [99 - 45(³√10)]/3⁵ = [11 - 5(³√10)]/27
Answer check:
{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(a - 1)/3]⁵ = [(√10 - 1)/3]⁵ = (1.154/3)⁵ = 0.385⁵ = 0.0084
[11 - 5(³√10)]/27 = [11 - 5(2.154)]/27 = 0.228/27 = 0.084; Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
{(³√100 - 1)/[3(³√10) + 3]}⁵ = [(³√10 - 1)/3]⁵ = [11 - 5(³√10)]/27