Problem 292 - Elementary Algebra

Question

In the following exercises, simplify. $$ (3-\sqrt{5})(3+\sqrt{5}) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is 4.

Step 1: Recognize the Identity

Identify the pattern in the given expression ewline ewline ewline ewline ewline ewline ewline ewline ewline The given expression is (3- sqrt{5})(3+ sqrt{5}).This fits the pattern of the difference of squares: (a-b)(a+b)= a^2-b^2.

Step 2: Apply the Difference of Squares

Apply the difference of squares formula: (a-b)(a+b)= a^2- b^2.In this case, a=3 and b= sqrt{5}.

Step 3: Compute the Squares

ewline Replace a and b with the values identified: a^2=3^2=9.Next, b^2=( sqrt{5})^2=5.ewline So, we have ewline 9- 5.

Step 4: Simplify the Result

Simplify the expression: 9-5=4.

Key Concepts

difference of squares

The difference of squares is a fundamental identity in algebra, perfect for simplifying certain types of expressions. The formula is given by \(a^2 - b^2 = (a - b)(a + b)\). This identity is useful when you have terms combined as a product of binomials. For example, the problem \( (3-\sqrt{5})(3+\sqrt{5})\) perfectly fits this pattern.
By recognizing that \(3 - \sqrt{5}\) and \(3 + \sqrt{5}\) are conjugates, you can apply the \(a^2 - b^2\) formula directly. Here, \(a = 3\) and \(b = \sqrt{5}\).
Following these steps, you get \( (3)^2 - (\sqrt{5})^2 = 9 - 5 = 4\). This illustrates the power of the difference of squares in simplifying polynomial expressions efficiently and effectively.

square roots

Square roots are the inverse operation of squaring a number. The square root function is denoted by \( \sqrt{} \). For any non-negative number \(x\), \( \sqrt{x}\) is a value that, when squared, gives \(x\) back. For example, \( \sqrt{9} = 3\) because \( 3^2 = 9\).
When dealing with algebraic expressions, you’ll often encounter numbers under the square root symbol. It’s crucial to remember that \( (\sqrt{a})^2 = a\). This property is essential when working with terms involving square roots, as it allows you to simplify expressions by removing the square root when squaring
In our exercise, \( (\sqrt{5})^2 = 5\), helping us transform the expression inside the difference of squares identity into a manageable form. As you practice, you'll find that understanding square roots simplifies many algebraic problems.

basic algebra

Basic algebra is the foundation of mathematics involving the manipulation of mathematical symbols. It introduces variables, expressions, equations, and the key properties of arithmetic operations. The goal is to use these skills to solve problems.
This exercise exemplifies basic algebra principles:

• Identify patterns and apply specific algebraic identities
• Simplify complex expressions
• Practice order of operations (exponents before subtraction)

Start by recognizing similar patterns in algebraic problems. Here, identifying the binomials fitting the difference of squares pattern quickly leads you to the solution:

• Apply the pattern \( (a-b)(a+b) = a^2 - b^2\)
• Compute the squares of terms \(a\) and \(b\)
• Simplify the result through basic arithmetic

By consistently using these steps, you’ll strengthen your algebraic skills and become more adept at solving a variety of mathematics problems efficiently.