find-the-product-of-roots-2021-1-3-x-log-2021-x-x-3-

Question Number 103047 by bemath last updated on 12/Jul/20

f i n d t h e p r o d u c t o f r o o t s

\sqrt[3]{2021} x^{\log_{2021} (x)} = x^{3}

Commented by mr W last updated on 12/Jul/20

Π x = 2021^{3} ?

Answered by floor(10²Eta[1]) last updated on 12/Jul/20

\log_{2021} (x) = y \Rightarrow x = 2021^{y}

\Rightarrow \sqrt[3]{2021} . {(2021^{y})}^{y} = {(2021^{y})}^{3}

2021^{1 / 3} . 2021^{y^{2}} = 2021^{3 y}

\Rightarrow y^{2} + \frac{1}{3} = 3 y \Rightarrow y^{2} - 3 y + \frac{1}{3} = 0

product = x_{1} . x_{2} = 2021^{y_{1}} . 2021^{y_{2}}

= 2021^{y_{1} + y_{2}} = 2021^{3}

Answered by bemath last updated on 14/Jul/20

s e t x = 2021^{y} \Leftrightarrow y = \log_{2021} (x)

\Rightarrow 2021^{\frac{1}{3}} {(2021^{y})}^{y} = 2021^{3 y}

2021^{y^{2} - 3 y + \frac{1}{3}} = 1

\Rightarrow y^{2} - 3 y + \frac{1}{3} = 0; Δ = 9 - \frac{4}{3} > 0

\Rightarrow (y - a) (y - b) = 0 \Rightarrow a + b = 3

A = 2021^{a}; B = 2021^{b}

\Leftrightarrow A . B = 2021^{a + b} = 2021^{3}

= 21^{3} (m o d 100) = 261 (m o d

1000)

Facebook Tweet Pin