Expanding ( frac{x}{x^6} ) Using the Binomial Theorem

When faced with the expression (frac{x}{x^6}), it is often useful to understand how to expand such an expression using the Binomial Theorem. This theorem provides a powerful method for expanding expressions of the form (a b)^n). In this case, we can use it to simplify the fraction.

Understanding the Binomial Theorem

The Binomial Theorem states that for any real numbers (a) and (b) and any non-negative integer (n): [ (a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k} b^k ] where (binom{n}{k}) is the binomial coefficient.

Applying the Binomial Theorem to ( frac{x}{x^6} )

Given the expression ( frac{x}{x^6} ), let's rewrite it in a form that allows us to apply the Binomial Theorem. Notice that this expression can be seen as a binomial expansion with (a x) and (b frac{1}{x}), and (n 6).

Step-by-Step Expansion

Using the Binomial Theorem:

[ x left(frac{1}{x}right)^6 sum_{k0}^{6} binom{6}{k} x^{6-k} left(frac{1}{x}right)^k ]

Each term in the sum can be calculated as follows:

For ( k 0 ): [ binom{6}{0} x^{6-0} left(frac{1}{x}right)^0 1 cdot x^6 x^6 ] For ( k 1 ): [ binom{6}{1} x^{6-1} left(frac{1}{x}right)^1 6 cdot x^5 cdot frac{1}{x} 6x^4 ] For ( k 2 ): [ binom{6}{2} x^{6-2} left(frac{1}{x}right)^2 15 cdot x^4 cdot frac{1}{x^2} 15x^2 ] For ( k 3 ): [ binom{6}{3} x^{6-3} left(frac{1}{x}right)^3 20 cdot x^3 cdot frac{1}{x^3} 20 ] For ( k 4 ): [ binom{6}{4} x^{6-4} left(frac{1}{x}right)^4 15 cdot x^2 cdot frac{1}{x^4} frac{15}{x^2} ] For ( k 5 ): [ binom{6}{5} x^{6-5} left(frac{1}{x}right)^5 6 cdot x^1 cdot frac{1}{x^5} frac{6}{x^4} ] For ( k 6 ): [ binom{6}{6} x^{6-6} left(frac{1}{x}right)^6 1 cdot x^0 cdot frac{1}{x^6} frac{1}{x^6} ]

Combining all the terms together, the expanded form of ( frac{x}{x^6} ) is:

[ x^6 - 6x^4 15x^2 - 20 frac{15}{x^2} - frac{6}{x^4} frac{1}{x^6} ]

Alternative Method: Pascal's Triangle

An alternative method to find the binomial coefficients is by using Pascal's Triangle. Here, the coefficients are taken from the 6th row of Pascal's Triangle (1, 6, 15, 20, 15, 6, 1) and applied to the powers of (x) and (frac{1}{x}).

The expanded form using Pascal's Triangle is identical to the one derived using the Binomial Theorem:

[ frac{x}{x^6} frac{x^6 - 6x^4 15x^2 - 20 15 cdot frac{1}{x^2} - 6 cdot frac{1}{x^4} frac{1}{x^6}}{x^6} ]

Conclusion

In conclusion, the expansion of ( frac{x}{x^6} ) using the Binomial Theorem or Pascal's Triangle leads to the simplified form:

[ boxed{x^6 - 6x^4 15x^2 - 20 frac{15}{x^2} - frac{6}{x^4} frac{1}{x^6}} ]

Understanding these methods can greatly enhance your algebraic manipulation skills, making complex expressions easier to handle.