Problems / Problem #4

4. Algebra

Hard Acceptance: 41.9%

If x + 1/x = 5, then what is the value of x³ + 1/x³?

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Algebra Quantitative Aptitude

Hard

If x + 1/x = 5, then what is the value of x³ + 1/x³?

Interactive Visual Explanation

Step 1 of 4

Step 1

Algebraic Expansion of Cubic Terms

We are given the linear sum equation. To find the sum of cubic terms, we must cube both sides of the original equation:

\left(x + \frac{1}{x}\right)^3 = 5^3

Step 2

Apply the Algebraic Identity

Use the expansion identity $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$. Applying this gives us:

x^3 + \frac{1}{x^3} + 3(x)\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right) = 125

Step 3

Substitute and Calculate

Since $x \cdot (1/x) = 1$ and we know that $x + 1/x = 5$, substitute these back into our equation:

x^3 + \frac{1}{x^3} + 3(1)(5) = 125 \implies x^3 + \frac{1}{x^3} + 15 = 125

Step 4

Final Subtraction

Subtract 15 from 125 to isolate the cubic sum term:

x^3 + \frac{1}{x^3} = 125 - 15 = 110

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CGL Exam Speed Trick

⚡ Shortcut Formula: For CGL, memorize this formula: If $x + 1/x = k$, then $x^3 + 1/x^3 = k^3 - 3k$. Since $k=5$, calculate $5^3 - 3(5) = 125 - 15 = 110$ in your head instantly.

⚡ Speed-Run CGL Cheatsheet

Mastering these core principles will let you solve similar questions under 30 seconds.

Core Identity:

If x + 1/x = k, then x³ + 1/x³ = k³ - 3k

Check the interactive visualizer in the first tab to see these formulas modeled geometrically!

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