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Question : 42 of 86

Marks: +1, -0

if

x=5^{\frac{1}{3}}+5^{-\frac{1}{3}}

5 x^{3}-15 x

is given by.

25
26
27
30 .

Solution:

Concept:

Use the expansion of

(a + a^{-1})^3

to simplify

x^3

in terms of

x

and the constant

5

.

Explanation:

Let

a = 5^{1/3}

, so

a^3 = 5

and

a^{-3} = 5^{-1} = \frac{1}{5}

.

Then

x = a + a^{-1}

.

Compute

x^3 = (a + a^{-1})^3 = a^3 + a^{-3} + 3(a + a^{-1}) = 5 + \frac{1}{5} + 3x = \frac{26}{5} + 3x

.

Multiply both sides by

5

:

5x^3 = 26 + 15x

.

Thus

5x^3 - 15x = 26

.

Answer:

5x^3 - 15x = 26

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