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Algebra and Applications Test 3

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Question : 9 of 25

Marks: +1, -0

If

x - \frac{1}{x} = 3

x^3 - \frac{1}{x^3}

is:

32
36
40
49

Solution:

Concept:

The difference of cubes identity connects

x-\frac{1}{x}

with

x^3-\frac{1}{x^3}

.

Explanation:

Use the identity

a^3-b^3=(a-b)^3+3ab(a-b)

.

Take

a=x

and

b=\frac{1}{x}

.

Then

ab=x\cdot\frac{1}{x}=1

.

Given

x-\frac{1}{x}=3

.

Substitute these values into the identity.

x^3-\frac{1}{x^3}=\left(x-\frac{1}{x}\right)^3+3\cdot1\cdot\left(x-\frac{1}{x}\right)

=3^3+3\cdot1\cdot3

=27+9

=36

This matches option B.

Answer:

The value of

x^3-\frac{1}{x^3}

is

36

.

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