Problem 138 - Elementary and Intermediate Algebra

Question

Explain why the product of \(\sqrt{m}+3\) and \(\sqrt{m}-3\) does not contain a radical.

Step-by-Step Solution

Answer

The product \((\sqrt{m} + 3)(\sqrt{m} - 3)\) is \(m - 9\), which has no radicals.

Step 1: Understanding the problem

We are given two expressions, \(\sqrt{m} + 3\) and \(\sqrt{m} - 3\), and asked to show why their product does not contain a radical. To do this, we need to perform the multiplication and check the resulting expression for radicals.

Step 2: Recognizing the pattern

Notice that \(\sqrt{m} + 3\) and \(\sqrt{m} - 3\) form a difference of squares, a special algebraic identity: \((a + b)(a - b) = a^2 - b^2\). In this case, \(a = \sqrt{m}\) and \(b = 3\).

Step 3: Applying the difference of squares

Using the difference of squares formula, we multiply \(\sqrt{m} + 3\) and \(\sqrt{m} - 3\):\[(\sqrt{m} + 3)(\sqrt{m} - 3) = (\sqrt{m})^2 - 3^2\]

Step 4: Simplifying the expression

Calculate each component of the expression:- \((\sqrt{m})^2 = m\)- \(3^2 = 9\)Thus, the product is:\[m - 9\]

Step 5: Conclusion

The resulting expression \(m - 9\) does not contain any radicals, as it is composed entirely of integers and the variable \(m\).

Key Concepts

Radicals in Algebra

Radicals in algebra often consist of expressions involving roots, like square roots. A radical usually appears in the form \( \sqrt{a} \), where \( a \) is some number or algebraic expression. When we see this symbol, it indicates that we must find a number which, when multiplied by itself, gives \( a \).
In algebra, radicals can sometimes complicate things, especially when they're part of a larger expression. That's why it's important to understand how to work with them effectively.
A crucial part of algebra is learning when and how to eliminate radicals. This process can help simplify expressions, making them easier to handle. In this specific problem, understanding how radicals interact, and recognizing patterns allows one to express the final result without any radicals at all.

Simplifying Expressions

To simplify expressions means to make them as clean and concise as possible. This involves reducing the expression to its simplest form.

In the case of the given problem, two binomials \( (\sqrt{m} + 3) \) and \( (\sqrt{m} - 3) \) are multiplied together. Simplifying this expression involves removing any unnecessary complexity, like radicals or like terms.
Ultimately, recognizing the pattern of the difference of squares makes it possible to simplify the expression from containing a radical to not containing one. Once simplified, the resulting expression is \( m - 9 \).

• Always look for patterns or identities that can simplify the work.
• Step-by-step simplification ensures that you don't miss important transformations.

Working through and understanding simplification can make more advanced operations much easier to match.

Algebraic Identities

Algebraic identities are formulas that provide a handy method to solve certain expressions or equations quickly. They are essentially shortcuts that are mathematically valid, allowing for simplification or expansion of expressions.

One key identity involved here is the difference of squares, which is written as \((a + b)(a - b) = a^2 - b^2\). This identity is extremely useful because it helps in instantly recognizing patterns where two terms are multiplied in the form of a sum and a difference.

Applying the identity to the provided expressions \( (\sqrt{m} + 3) \) and \( (\sqrt{m} - 3) \), we set \( a = \sqrt{m} \) and \( b = 3 \). Hence, following the identity, the product immediately simplifies to \((\sqrt{m})^2 - 3^2 = m - 9\). Consequently, this prompts the removal of the radical leaving a straightforward expression.
Practicing with and understanding algebraic identities can dramatically reduce the work needed by performing simple, yet profound transformations.