जर \(a + \frac{1}{a} = 5\) , तर \({a^3} + \frac{1}{{{a^3}}}\) चे मूल्य काय आहे?

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Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 12 Dec 2022 Shift 2)

Answer 1

दिलेल्याप्रमाणे:

\(a + \frac{1}{a} = 5\)\(a + \frac{1}{a} = 5\)

वापरलेली संकल्पना:

(a + b)3 = a3 + 3ab(a + b) + b3

गणना:

\(a + \frac{1}{a} = 5\)\(a + \frac{1}{a} = 5\)

⇒ \(a + \frac{1}{a} = 5\)\((a + \frac{1}{a})^3 = 5^3\)

⇒ \(a^3 + \frac{1}{a^{3}} + 3(a + \frac {1}{a}) = 5^3\)\(a^3 + \frac{1}{a^{3}} + 3(a + \frac {1}{a}) = 5^3\)

⇒ \(a^3 + \frac{1}{a^{3}} + 3 \times 5 = 125\)\(a^3 + \frac{1}{a^{3}} + 3 \times 5 = 125\)

⇒ \(a^3 + \frac{1}{a^{3}} = 125 - 15\)\(a^3 + \frac{1}{a^{3}} = 125 - 15\)

⇒ \(a^3 + \frac{1}{a^{3}} = 110\)\(a^3 + \frac{1}{a^{3}} = 110\)

∴ \({a^3} + \frac{1}{{{a^3}}}\) चे मूल्य 110 आहे.