Problem 58 - Algebra 2

Question

Which expression is NOT equal to 13\(?\) A. \((4+\sqrt{3})(4-\sqrt{3})\) C. \((6+\sqrt{23})(6-\sqrt{23})\) B. \((5-2 \sqrt{3})(5+2 \sqrt{3})\) D. \((7-\sqrt{6})(7+\sqrt{6})\)

Step-by-Step Solution

Answer

Option D is not equal to 13.

Step 1: Simplify Choice A

Simplify the expression \((4+\sqrt{3})(4-\sqrt{3})\). This is in the form of \((a+b)(a-b) = a^2 - b^2\), therefore the simplified expression is \((4)^2 - (\sqrt{3})^2 = 16 - 3 = 13\). So, Choice A equals to 13

Step 2: Simplify Choice C

Next, simplify the expression \((6+\sqrt{23})(6-\sqrt{23})\). By applying the same property of \(a^2 - b^2\), it becomes \((6)^2 - (\sqrt{23})^2 = 36 - 23 = 13\). So, Choice C equals to 13

Step 3: Simplify Choice B

Now, simplify the expression \((5-2 \sqrt{3})(5+2 \sqrt{3})\). Applying the formula, it becomes \((5)^2 - (2 \sqrt{3})^2 = 25 - 12 = 13\). So, Choice B equals to 13

Step 4: Simplify Choice D

Lastly, simplify the expression \((7-\sqrt{6})(7+\sqrt{6})\). Applying the formula, it becomes \((7)^2 - (\sqrt{6})^2 = 49 - 6 = 43\). So, Choice D does not equal to 13 but 43

Key Concepts

Difference of Squares

The difference of squares is a commonly used algebraic identity that helps simplify expressions of the form \((a+b)(a-b)\). This pattern results in \(a^2 - b^2\). Let's consider why this happens. When we expand \((a+b)(a-b)\), we use distributive property:

• First, multiply \(a\) by \(a\) to get \(a^2\).
• Then, multiply \(a\) by \(-b\) to get \(-ab\).
• Next, multiply \(b\) by \(a\) to get \(+ab\).
• Finally, multiply \(b\) by \(-b\) to get \(-b^2\).

Notice that the \(-ab\) and \(+ab\) cancel each other out, leaving us with \(a^2 - b^2\). This identity is powerful because it converts products into differences, simplifying complex algebraic expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Consider it a way to represent numbers using symbols and letters. It can be as simple as \( x + 2 \) or as complex as \((3x^2 + 2x - 5)\). Let's break down the components:

• Terms: Parts of the expression separated by '+' or '-' signs.
• Variables: Symbols (often letters) that stand for unknown values.
• Constants: Numbers without variables, like \(2\) or \(5\).
• Coefficients: Numbers that multiply a variable, such as \(3\) in \(3x\).

Understanding how to manipulate and simplify these expressions is crucial, especially when applying operations like addition, subtraction, and factorization.

Radicals

Radicals are expressions that include the root symbol \(\sqrt{}\), which indicates a root of a number. Commonly, you’ll encounter square roots, like \(\sqrt{4}\). Here are some essential points about radicals:

• Square Roots: The most common type is the square root, written as \(\sqrt{x}\), which represents a number that, when multiplied by itself, gives \(x\).
• Radical Simplification: This involves rewriting the radical in its simplest form, such as \(\sqrt{18} = 3\sqrt{2}\).
• Radical Multiplication: Treat radicals like variables when multiplying: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

Understanding radicals is vital because many algebraic expressions include them, requiring simplification for easier problem-solving.

Equivalent Expressions

Equivalent expressions are expressions that, although they may look different, have the same value. This concept allows you to rewrite expressions in different forms. For instance, consider \((4+\sqrt{3})(4-\sqrt{3})\) and \(13\). After simplifying using the difference of squares, both expressions equal \(13\).

• Why Are These Important? Because they help in simplifying and solving algebraic equations by transforming complex expressions into more manageable forms.
• Verification: To verify if two expressions are equivalent, simplify each expression as much as possible. If they simplify to the same value, they are equivalent.
• Properties: Equivalent expressions maintain equality even when altered through addition, subtraction, multiplication, or division, provided the operations are performed on both sides equally.

Knowing how to identify and work with equivalent expressions is essential for success in algebra.