If $${x^2} + \frac{1}{{{x^2}}} = \frac{7}{4}$$ for x > 0 then what is the value of $${x^3} + \frac{1}{{{x^3}}} = ?$$
A. $$\frac{{3\sqrt 3 }}{5}$$
B. $$\frac{{3\sqrt {15} }}{5}$$
C. $$\frac{{3\sqrt {15} }}{8}$$
D. $$\frac{{3\sqrt 5 }}{8}$$
Answer: Option C
Solution(By Examveda Team)
$$\eqalign{ & {\text{Given, }}{x^2} + \frac{1}{{{x^2}}} = \frac{7}{4} \cr & {\text{Adding 2 to both sides}} \cr & {x^2} + \frac{1}{{{x^2}}} + 2 = \frac{7}{4} + 2 \cr & {\left( {x + \frac{1}{x}} \right)^2} = \frac{{15}}{4} \cr & x + \frac{1}{x} = \frac{{\sqrt {15} }}{2} \cr & \because \,{x^3} + \frac{1}{{{x^3}}} = {\left( {x + \frac{1}{x}} \right)^3} - 3\left( {x + \frac{1}{x}} \right) \cr & \therefore \,{x^3} + \frac{1}{{{x^3}}} = \frac{{15 \times \sqrt {15} }}{8} - 3 \times \frac{{\sqrt {15} }}{2} \cr & = \frac{{15\sqrt {15} - 12\sqrt {15} }}{8} \cr & = \frac{{3\sqrt {15} }}{8} \cr} $$Related Questions on Algebra
If $$p \times q = p + q + \frac{p}{q}{\text{,}}$$ then the value of 8 × 2 is?
A. 6
B. 10
C. 14
D. 16
A. $$1 + \frac{1}{{x + 4}}$$
B. x + 4
C. $$\frac{1}{x}$$
D. $$\frac{{x + 4}}{x}$$
A. $$\frac{{20}}{{27}}$$
B. $$\frac{{27}}{{20}}$$
C. $$\frac{6}{8}$$
D. $$\frac{8}{6}$$
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