If \(x+\dfrac{1}{x}=2\sqrt{3}\), then find the value of \(x^3+\dfrac{1}{x^3}\).
\(18\sqrt{3}\)
To find the value of \(x^3+\dfrac{1}{x^3}\) given that \(x+\dfrac{1}{x}=2\sqrt{3}\), we can utilize algebraic identities. Let's solve this problem step by step.
- Start with the identity for cubes: \(x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right)^3 - 3\left(x + \dfrac{1}{x}\right)\).
- Substitute the given value: \(x + \dfrac{1}{x} = 2\sqrt{3}\).
- Calculate \((x + \dfrac{1}{x})^3\):
- \((2\sqrt{3})^3 = 8 \times 3^{3/2} = 8 \times 3\sqrt{3} = 24\sqrt{3}\).
- Find \(3 \left(x + \dfrac{1}{x}\right)\):
- \(3 \times 2\sqrt{3} = 6\sqrt{3}\).
- Substitute into the identity: \(x^3 + \dfrac{1}{x^3} = 24\sqrt{3} - 6\sqrt{3}\).
- Simplify: \(x^3 + \dfrac{1}{x^3} = 18\sqrt{3}\).
Thus, the value of \(x^3+\dfrac{1}{x^3}\) is \(18\sqrt{3}\), which corresponds to the correct option.
Similar Questions
\(27^3 + 10^3 - 29^3 + 246\) is equal to:
What is the result of \(\dfrac{0.5^3 + 0.1^3 - 0.6^3}{3 \times 0.5 \times 0.1 \times 0.6}\)?
- The rate of simple interest per annum at which a sum of money double itself in 16 2/3 yrs. is
- Given, $x = \sqrt{3}$, what is $(x + 1)^2 + (x - 1)^2$?
Important Questions from Identities
Simplify.
\(\frac{2.5 \times 2.5 \times 2.5-1.5 \times 1.5 \times 1.5}{2.5 \times 2.5+2.5 \times 1.5+1.5 \times 1.5}\)
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
