Expanding Binomials with Pascal’s Triangle: A Simplified Approach

When faced with expressions like (sqrt{a}sqrt{b}^8), the initial reaction may be to seek a solution. However, in the world of mathematics, we don’t ‘solve’ these types of expressions; instead, we ‘expand’ them. This article will explore how to expand such expressions using the powerful tool known as Pascal’s Triangle.

What is Pascal's Triangle?

Pascal's Triangle is a triangular array of the binomial coefficients. Each number is the sum of the two numbers directly above it. The triangle starts with a single 1 at the top, followed by a row of 1s, and each subsequent row is constructed by adding the two numbers above it. The coefficients of the expansion of a binomial are found in each row of Pascal's Triangle. Here is an example of Pascal's Triangle up to the 8th row:

t tPascal’s Triangle up to the 8th Row

Let's break down the process step by step.

Step 1: Understanding Binomial Expansions

When expanding a binomial like ((sqrt{a}(sqrt{b})^8)), we use the Binomial Theorem, which states that:

((x y)^n sum_{k0}^{n} binom{n}{k} x^{n-k} y^k)

Where (binom{n}{k}) is the binomial coefficient, which can be found in the (n)th row of Pascal's Triangle.

Step 2: Construct Pascal’s Triangle

Since the binomial power is 8, we construct Pascal's Triangle up to the 8th row (remember, the 0th row is the first row).

t tPascal’s Triangle up to the 8th Row

The coefficients for the 8th row are: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Step 3: Expanding the Binomial

We will now expand the binomial ((sqrt{a}(sqrt{b})^8)) using the coefficients from Pascal's Triangle. The expanded form will be:

(sqrt{a}sqrt{b}^8 (sqrt{a})^8 8(sqrt{a})(sqrt{b})^7 28(sqrt{a})^6(sqrt{b})^2 56(sqrt{a})^5(sqrt{b})^3 70(sqrt{a})^4(sqrt{b})^4 56(sqrt{a})^3(sqrt{b})^5 28(sqrt{a})^2(sqrt{b})^6 8(sqrt{a})(sqrt{b})^7 (sqrt{b})^8)

Now, let's simplify each term:

t(((sqrt{a})^8 a^4) t(8(sqrt{a})(sqrt{b})^7 8a^{0.5}b^{3.5}) t(28(sqrt{a})^6(sqrt{b})^2 28a^3b) t(56(sqrt{a})^5(sqrt{b})^3 56a^{2.5}b^{3.5}) t(70(sqrt{a})^4(sqrt{b})^4 70a^2b^2) t(56(sqrt{a})^3(sqrt{b})^5 56a^{1.5}b^{2.5}) t(28(sqrt{a})^2(sqrt{b})^6 28ab^3) t(8(sqrt{a})(sqrt{b})^7 8a^{0.5}b^{3.5}) t(((sqrt{b})^8 b^4)

Notice the symmetry in the simplified terms. Each term reflects the corresponding coefficient from the 8th row of Pascal's Triangle, adjusted for the powers of (sqrt{a}) and (sqrt{b}).

Constructing Pascal's Triangle and Expanding the Binomial

To construct Pascal's Triangle, we start with the 0th row (which is just 1). Each subsequent row is constructed by summing the two numbers directly above it. For the 8th row, we use the coefficients 1, 8, 28, 56, 70, 56, 28, 8, 1. These coefficients are then applied to the expanding binomial:

Step 1: Expand Binomial with Each Term in Descending Ascending Order of Powers

Let's expand ((sqrt{a}(sqrt{b})^8)):

tWrite the powers of (sqrt{a}) and (sqrt{b}) in descending and ascending orders respectively: t(sqrt{a}sqrt{b}^8 sqrt{a}^8sqrt{b}^0 8sqrt{a}^7sqrt{b}^1 28sqrt{a}^6sqrt{b}^2 56sqrt{a}^5sqrt{b}^3 70sqrt{a}^4sqrt{b}^4 56sqrt{a}^3sqrt{b}^5 28sqrt{a}^2sqrt{b}^6 8sqrt{a}^1sqrt{b}^7 sqrt{a}^0sqrt{b}^8)

Step 2: Simplify Each Term, Insert the Coefficients

Now, simplify each term by inserting the coefficients from the 8th row of Pascal's Triangle:

t(sqrt{a}^8sqrt{b}^0 a^4) t(8sqrt{a}^7sqrt{b}^1 8a^{3.5}b^{0.5}) t(28sqrt{a}^6sqrt{b}^2 28a^3b^1) t(56sqrt{a}^5sqrt{b}^3 56a^{2.5}b^{1.5}) t(70sqrt{a}^4sqrt{b}^4 70a^2b^2) t(56sqrt{a}^3sqrt{b}^5 56a^{1.5}b^{2.5}) t(28sqrt{a}^2sqrt{b}^6 28ab^3) t(8sqrt{a}^1sqrt{b}^7 8a^{0.5}b^{3.5}) t(sqrt{a}^0sqrt{b}^8 b^4)

Conclusion

By understanding and utilizing Pascal's Triangle, we can greatly simplify the process of expanding binomials. This method not only provides a clear and systematic approach but also helps in understanding the underlying mathematical principles at play.

Getting familiar with the Binomial Theorem and Pascal's Triangle early in your mathematical education can greatly enhance your problem-solving skills and provide a foundation for more complex mathematical concepts.

If you have any further questions or need more detailed explanations, feel free to ask. Happy learning!