Problem 290 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((9-\sqrt{6})^{2}\) (b) \((10+3 \sqrt{7})^{2}\)

Step-by-Step Solution

Answer

For part (a): \( 87 - 18 \sqrt{6} \). For part (b): \( 163 + 60 \sqrt{7} \).

Step 1: Expand using the binomial theorem

Use the formula \( a^2 - 2ab + b^2 \) to expand \( (a - b)^2 \). For \( (9 - \sqrt{6})^2 \), let \( a = 9 \) and \( b = \sqrt{6} \).

Step 2: Apply the formula

Calculate \( 9^2 - 2 \times 9 \times \sqrt{6} + (\sqrt{6})^2 \). This results in \( 81 - 18 \sqrt{6} + 6 \).

Step 3: Combine like terms

Combine the constants: \( 81 + 6 \). The expression simplifies to \( 87 - 18 \sqrt{6} \).

Step 4: Expand using the binomial theorem (part b)

Use the formula \( a^2 + 2ab + b^2 \) to expand \( (a + b)^2 \). For \( (10 + 3 \sqrt{7})^2 \), let \( a = 10 \) and \( b = 3 \sqrt{7} \).

Step 5: Apply the formula (part b)

Calculate \( 10^2 + 2 \times 10 \times 3 \sqrt{7} + (3 \sqrt{7})^2 \). This results in \( 100 + 60 \sqrt{7} + 63 \).

Step 6: Combine like terms (part b)

Combine the constants: \( 100 + 63 \). The expression simplifies to \( 163 + 60 \sqrt{7} \).

Key Concepts

Binomial Theorem

The binomial theorem is a key principle in algebra for expanding expressions that are raised to a power. It helps in simplifying expressions involving binomials, which are sums or differences of two terms. Using the binomial theorem, we can expand expressions like \( (a+b)^2 \) or \( (a-b)^2 \) effortlessly. For example, in the given exercise, we use the formulas:

• \[ (a - b)^2 = a^2 - 2ab + b^2 \]
• \[ (a + b)^2 = a^2 + 2ab + b^2 \]

These formulas simplify the expansion by breaking them down into manageable parts, making it easier to work with. For instance, for \( (9 - \sqrt{6})^2 \), we substitute 'a' with 9 and 'b' with \sqrt{6}, then apply the formula to get the expanded form.

Combine Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. When expanding expressions, you often get terms that are similar and can be combined to form a simpler expression. Like terms are terms that have identical variable parts. In the example \( 81 - 18 \sqrt{6} + 6 \), you can combine the constants: 81 and 6. This process reduces complexity and makes it easier to interpret the expression. Therefore, the expression simplifies to \( 87 - 18 \sqrt{6} \). A similar process is used in part (b) of our exercise: \( 100 + 60 \sqrt{7} + 63 \), where we combine constants 100 and 63 for a final simplified form: \( 163 + 60 \sqrt{7} \).

Expand Expressions

Expanding expressions is a technique where a compact form of an expression is expanded into a more detailed form. This is particularly useful in algebra where binomial expressions need to be expanded for simplification or further calculation. Using the binomial theorem, we expand expressions step by step. Take \( (10 + 3 \sqrt{7})^2 \) as an example. By setting 'a' to 10 and 'b' to 3 \sqrt{7}, and then using the binomial theorem formula \[ a^2 + 2ab + b^2 \], we can methodically expand and simplify the expression. This results in detailed steps that break down the expression into smaller, easily manageable components, making the expression \[ 100 + 60 \sqrt{7} + 63 \], which simplifies further to \[ 163 + 60 \sqrt{7} \]. Each step is logical, ensuring that the expanded expression is clear and precise.