If X=7-4√3 then find the value of x³+1/x³

Given that \( x = 7 - 4\sqrt{3} \), we are tasked with finding the value of \( x^3 + \frac{1}{x^3} \).

### Step 1: Derive \( x + \frac{1}{x} \)
First, we find \( x + \frac{1}{x} \). Multiply \( x = 7 - 4\sqrt{3} \) by its conjugate:

\[
x \cdot \left( \frac{1}{x} \right) = 1
\]

The conjugate of \( 7 - 4\sqrt{3} \) is \( 7 + 4\sqrt{3} \), so:

\[
x \cdot \left( \frac{1}{x} \right) = \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} = 1
\]

Now multiply the numerator and denominator by \( 7 + 4\sqrt{3} \):

\[
x + \frac{1}{x} = \frac{7 + 4\sqrt{3}}{

If X = 7 - 4√3 then find the value of x³ + 1/x³

Solve the problem of finding x³ + 1/x³ when x = 7 - 4√3 using conjugates and algebraic identities. This step-by-step guide explains the use of conjugates and cubic expansions, suitable for students in Grades 10-12.

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