Problem 298 - Elementary Algebra

Question

In the following exercises, simplify. $$ (12-5 \sqrt{5})(12+5 \sqrt{5}) $$

Step-by-Step Solution

Answer

The simplified result is 19.

Step 1 - Recognize the pattern

This expression matches the form of a difference of squares: \(a^2 - b^2 = (a-b)(a+b)\). Identify \(a\) and \(b\) from the given terms. For \(a = 12\) and \(b = 5 \sqrt{5}\).

Step 2 - Apply the difference of squares formula

Using the difference of squares formula \(a^2 - b^2\), calculate \(12^2\) for \(a^2\) and \( (5 \sqrt{5})^2\) for \(b^2\).

Step 3 - Calculate \(a^2\)

Calculate \(12^2\). This equals \(144\).

Step 4 - Calculate \( (5 \sqrt{5})^2\)

Calculate \( (5 \sqrt{5})^2\). This equals \( 5^2 \cdot (\sqrt{5})^2 = 25 \cdot 5 = 125 \).

Step 5 - Subtract the squares

Subtract \(b^2\) from \(a^2\). Thus, \(144 - 125 = 19\).

Key Concepts

difference of squares

The difference of squares is a special algebraic pattern that helps simplify expressions. It is represented by the formula \(a^2 - b^2 = (a-b)(a+b)\). This means that if you have two terms, one subtracted from another, you can factor them into a product of a sum and a difference.

For example, in the expression \((12 - 5 \sqrt{5})(12 + 5 \sqrt{5})\), we recognize it as a difference of squares where:

• \(a = 12\)
• \(b = 5 \sqrt{5}\)

Applying the difference of squares formula, we have: \(a^2 - b^2\). This allows us to break it down into simpler components.

The main thing to remember is to identify your \(a\) and \(b\) clearly and then use the formula accurately.

radicals

Understanding radicals is crucial when dealing with algebraic expressions involving square roots. A radical is an expression that includes a root symbol. The square root of a number \(x\) is represented by \(\sqrt{x}\).

For instance, in our original expression, \(b\) is identified as \(5 \sqrt{5}\). When squaring this term \((5 \sqrt{5})^2\), we follow these steps:

• Square the coefficient: \(5^2 = 25\)
• Square the radical: \((\sqrt{5})^2 = 5\)

Combining these, we get \(25 \cdot 5 = 125\).

This tells you that squared radicals should be handled carefully, ensuring each part is squared correctly.

polynomial multiplication

Polynomial multiplication extends the concept of multiplying simple algebraic expressions to more complex forms. It involves multiplying each term in one polynomial by each term in another polynomial.

Here’s a quick overview: Given two binomials \((a+b)\) and \((a-b)\), their multiplication is structured as:

• \(a \cdot a = a^2\)
• \(a \cdot (-b) = -ab\)
• \(b \cdot a = ab\)
• \(b \cdot (-b) = -b^2\)

Simplifying this gives us \(a^2 - b^2\).

In our given problem, these steps are mirrored in multiplying \((12 - 5 \sqrt{5})(12 + 5 \sqrt{5})\). Recognizing the multiplication pattern is crucial to solving the problem effectively. The result of the polynomial multiplication fits back into the difference of squares formula: \(144 - 125 = 19\). This makes the solving process much smoother and the result more comprehensible.