Question

answer the question
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Answer 1

$\mathsf{Given : \dfrac{\sqrt[3]{0.512}}{x} = \sqrt[3]{1000000}}$

$\mathsf{\implies x = \dfrac{\sqrt[3]{0.512}}{\sqrt[3]{1000000}}}$

$\textsf{We know that :}$

$\mathsf{\bigstar\;\;0.512 = 0.8 \times 0.8 \times 0.8 = (0.8)^3}$

$\mathsf{\bigstar\;\;1000000 = 100 \times 100 \times 100 = (100)^3}$

$\mathsf{\implies x = \dfrac{\sqrt[3]{(0.8)^3}}{\sqrt[3]{(100)^3}}}$

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{\sqrt[n]{a} = a^{\frac{1}{n}}}}}$

$\mathsf{\implies x = \dfrac{[(0.8)^3]^{\frac{1}{3}}}{[(100)^3]^{\frac{1}{3}}}}$

$\mathsf{\implies x = \dfrac{[0.8]^{\frac{3}{3}}}{[100]^{\frac{3}{3}}}}$

$\mathsf{\implies x = \dfrac{0.8}{100}}$

$\mathsf{\implies x = 0.008}$

Answer 2

$\huge{\mathsf{\star{\underline{Concept\: behind:-}}}}$

$\sf\textsf{The\: basic concepts of exponents and radicals only.}$

$\mathsf{It \:is \:given\: that :-{\implies{\dfrac{\sqrt[3]{0.512}}{x} = \sqrt[3]{1000000}}}}$

$\mathsf{\implies x = \dfrac{\sqrt[3]{0.512}}{\sqrt[3]{1000000}}}$

$\mathsf{\implies x = \dfrac{\sqrt[3]{(0.8)^3}}{\sqrt[3]{(100)^3}}}$

$\mathsf{\implies x = \dfrac{[(0.8)^3]^{\frac{1}{3}}}{[(100)^3]^{\frac{1}{3}}}}$

$\mathsf{\implies x = \dfrac{[0.8]^{\frac{3}{3}}}{[100]^{\frac{3}{3}}}}$

$\mathsf{\implies x = \dfrac{0.8}{100}}$

$\mathsf{\implies{\boxed{\star{\bold{\underline{x = 0.008}}}}}}$

$\mathsf{So,\:x=0.008}$