Ifx-21-3-22-3-Show-that-x3-6x-6-

X=2^1/3+2^2/3 Cubing both sides, x^3= (2^1/3+2^2/3 )^3=2+2^2+3× 2^1/3× 2^2/3 (2^1/3+2^2/3 ) ⇒ x^3=2+4+3× 2^1/3+ 2/3× x ⇒ x^3=6+3×2 × x ⇒ x^3=6x+6 ⇒∴ x^3 6x=6 He ...

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Standard IX

Mathematics

Laws of Exponents for Real Numbers

Ifx=21/3+22/3...

Question

If $x = 2^{\frac{1}{3}} + 2^{\frac{2}{3}}, S h o w t h a t x^{3} - 6 x = 6.$

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Solution

$x = 2^{\frac{1}{3}} + 2^{\frac{2}{3}} C u b i n g b o t h s i d e s, x^{3} = {(2^{\frac{1}{3}} + 2^{\frac{2}{3}})}^{3} = 2 + 2^{2} + 3 \times 2^{\frac{1}{3}} \times 2^{\frac{2}{3}} (2^{\frac{1}{3}} + 2^{\frac{2}{3}}) \Rightarrow x^{3} = 2 + 4 + 3 \times 2^{\frac{1}{3} + \frac{2}{3}} \times x \Rightarrow x^{3} = 6 + 3 \times 2 \times x \Rightarrow x^{3} = 6 x + 6 \Rightarrow∴ x^{3} - 6 x = 6 H e n c e P r o v e d .$

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Ifx=213+223,Show that x3−6x=6.

x=213+223Cubing both sides,x3=(213+223)3=2+22+3×213×223(213+223)⇒x3=2+4+3×213+23×x⇒x3=6+3×2×x⇒x3=6x+6⇒∴x3−6x=6 Hence Proved.

If $x = 2^{\frac{1}{3}} + 2^{\frac{2}{3}}, S h o w t h a t x^{3} - 6 x = 6.$