Apply binomial expansion is a crucial mathematical tool for Science Laboratory Technology professionals in Kenya. It allows approximation of roots and powers of expressions, which is essential in error estimation and data analysis in laboratory measurements. Mastery of this topic enables precise calculation of small changes, improving accuracy in experimental results and quality control in scientific processes.

3.1 Roots of numbers using binomial theorem expansions

The binomial theorem provides a way to expand expressions raised to any power, including fractional powers, which is useful for calculating roots of numbers. In laboratory technology, this method helps approximate square roots or cube roots when exact values are difficult to compute directly, such as during chemical concentration calculations or instrument calibration.

3.1.1 Pascal’s triangle

Pascal’s triangle is a triangular array of binomial coefficients that simplifies the expansion of binomial expressions. Each row corresponds to coefficients in the expansion of \( (a + b)^n \), where \( n \) is a non-negative integer. Understanding Pascal’s triangle is foundational for applying the binomial theorem efficiently.

$$ \text{Row } n: \binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n} $$

Worked Examples

Example 1: Find the coefficients for \( (a + b)^4 \) using Pascal’s triangle.

Given: \( n = 4 \)

Row 4 of Pascal’s triangle is:

$$ \binom{4}{0} = 1, \quad \binom{4}{1} = 4, \quad \binom{4}{2} = 6, \quad \binom{4}{3} = 4, \quad \binom{4}{4} = 1 $$

Answer: 1, 4, 6, 4, 1

Example 2: Use Pascal’s triangle to write the expansion of \( (x + y)^3 \).

Given: \( n = 3 \)

Coefficients: \( 1, 3, 3, 1 \)

Expansion:

$$ (x + y)^3 = 1 \cdot x^3 + 3 \cdot x^2 y + 3 \cdot x y^2 + 1 \cdot y^3 $$

Answer: \( x^3 + 3x^2 y + 3x y^2 + y^3 \)

Example 3: Determine the coefficient of \( a^2 b^3 \) in \( (a + b)^5 \).

Given: \( n = 5 \), powers of \( a \) and \( b \) are 2 and 3 respectively.

Coefficient:

$$ \binom{5}{2} = \frac{5!}{2! (5-2)!} = \frac{120}{2 \times 6} = 10 $$

Answer: The coefficient is 10

3.1.2 Binomial expansion for fractional powers

The binomial theorem can be extended to fractional powers using the generalized binomial series. For \( |x| < 1 \),

$$ (1 + x)^n = 1 + n x + \frac{n (n - 1)}{2!} x^2 + \frac{n (n - 1) (n - 2)}{3!} x^3 + \cdots $$

where \( n \) is a fraction or negative number. This is useful for approximating roots such as square roots (\( n = \frac{1}{2} \)) or cube roots (\( n = \frac{1}{3} \)).

Worked Examples

Example 1: Approximate \( \sqrt{1.04} \) using binomial expansion up to the second order term.

Given: \( n = \frac{1}{2} \), \( x = 0.04 \)

Formula:

$$ (1 + x)^{\frac{1}{2}} = 1 + \frac{1}{2} x + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} x^2 $$

Substitution:

$$ = 1 + \frac{1}{2} \times 0.04 + \frac{\frac{1}{2} \times (-\frac{1}{2})}{2} \times (0.04)^2 $$

Calculations:

$$ = 1 + 0.02 - \frac{1}{4} \times \frac{1}{2} \times 0.0016 $$

$$ = 1 + 0.02 - 0.0002 $$

Answer: 1.0198

Example 2: Approximate \( (1 - 0.05)^{\frac{1}{3}} \) up to the third order term.

Given: \( n = \frac{1}{3} \), \( x = -0.05 \)

Formula:

$$ (1 + x)^n = 1 + n x + \frac{n (n - 1)}{2} x^2 + \frac{n (n - 1) (n - 2)}{6} x^3 $$

Substitution:

$$ = 1 + \frac{1}{3} (-0.05) + \frac{\frac{1}{3} (\frac{1}{3} - 1)}{2} ( -0.05)^2 + \frac{\frac{1}{3} (\frac{1}{3} - 1) (\frac{1}{3} - 2)}{6} (-0.05)^3 $$

Calculations:

$$ = 1 - 0.0167 + \frac{\frac{1}{3} \times (-\frac{2}{3})}{2} \times 0.0025 + \frac{\frac{1}{3} \times (-\frac{2}{3}) \times (-\frac{5}{3})}{6} \times (-0.000125) $$

$$ = 1 - 0.0167 - 0.00028 - 0.000023 $$

$$ = 0.9830 $$

Answer: 0.9830

Example 3: Find the approximate cube root of 1.1 using binomial expansion up to the second order term.

Given: \( n = \frac{1}{3} \), \( x = 0.1 \)

Formula:

$$ (1 + x)^n = 1 + n x + \frac{n (n - 1)}{2} x^2 $$

Substitution:

$$ = 1 + \frac{1}{3} \times 0.1 + \frac{\frac{1}{3} (\frac{1}{3} - 1)}{2} \times (0.1)^2 $$

Calculations:

$$ = 1 + 0.0333 + \frac{\frac{1}{3} \times (-\frac{2}{3})}{2} \times 0.01 $$

$$ = 1 + 0.0333 - 0.0011 $$

Answer: 1.0322

3.1.3 Application of binomial expansion to roots in laboratory calculations

In Science Laboratory Technology, binomial expansion approximates roots in concentration calculations, dilution factors, and error estimations. This approach is practical when laboratory equipment provides approximate readings and exact root calculations are time-consuming.

Worked Examples

Example 1: Approximate the square root of a measured absorbance value of 1.09.

Given: \( n = \frac{1}{2} \), \( x = 0.09 \)

Formula:

$$ (1 + x)^{\frac{1}{2}} = 1 + \frac{1}{2} x + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} x^2 $$

Substitution:

$$ = 1 + 0.5 \times 0.09 + \frac{0.5 \times (-0.5)}{2} \times (0.09)^2 $$

Calculations:

$$ = 1 + 0.045 - 0.0010125 $$

Answer: 1.04399

Example 2: Approximate the cube root of a reagent volume change from 1 L to 1.08 L.

Given: \( n = \frac{1}{3} \), \( x = 0.08 \)

Formula:

$$ (1 + x)^{\frac{1}{3}} = 1 + \frac{1}{3} x + \frac{\frac{1}{3} (\frac{1}{3} - 1)}{2} x^2 $$

Substitution:

$$ = 1 + 0.3333 \times 0.08 + \frac{0.3333 \times (-0.6667)}{2} \times (0.08)^2 $$

Calculations:

$$ = 1 + 0.0267 - 0.000711 $$

Answer: 1.02599

Example 3: Approximate \( \sqrt{0.96} \) for a pH buffer concentration calculation.

Given: \( n = \frac{1}{2} \), \( x = -0.04 \)

Formula:

$$ (1 + x)^{\frac{1}{2}} = 1 + \frac{1}{2} x + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} x^2 $$

Substitution:

$$ = 1 - 0.02 + \frac{0.5 \times (-0.5)}{2} \times 0.0016 $$

Calculations:

$$ = 1 - 0.02 - 0.0002 $$

Answer: 0.9798

Chapter Summary

This chapter focused on applying the binomial expansion to solve various mathematical problems relevant to science and engineering. It began with finding roots of numbers using binomial theorem expansions, highlighting the role of Pascal's triangle in determining the coefficients for the expansion terms. The chapter then explored how small changes affect calculations by introducing errors, categorizing them into absolute, relative, and percentage errors, and showing how the binomial theorem can estimate these errors effectively. Finally, the chapter covered the fundamental concepts of permutation and combination, explaining how to count arrangements and selections in different scenarios. Through these topics, students developed a strong foundation in binomial expansion applications, error analysis, and combinatorial mathematics, essential for solving practical problems in technical fields.