Problem 90 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Which of the following does not equal \(\sqrt{48} ?\) $$ \text { (A) } \sqrt{2} \cdot \sqrt{24} $$ $$ \text { (B)} 2 \sqrt{12} $$ $$ \text { (C) } 4 \sqrt{3} $$ $$ \text { (D) } 12 \sqrt{16} $$

Step-by-Step Solution

Answer

The option that does not equal to \(\sqrt{48}\) is (D) \(12\sqrt{16}\)

Step 1: Calculate the Square root of 48

First, calculate the square root of 48 which equals to \(4 \sqrt{3} \) using simplification methods.

Step 2: Compare with options

Next, compare the result with given multiple choices. Choice (A): \( \sqrt{2} \cdot \sqrt{24} = \sqrt{48} = 4\sqrt{3} \). So choice A is equivalent to \(\sqrt{48}\).Choice (B): \(2\sqrt{12} = 2(2\sqrt{3}) = 4\sqrt{3}\). So choice B is also equivalent.Choice (C): \(4\sqrt{3} = 4\sqrt{3}\). Thus choice C is equivalent again.Choice (D): \(12\sqrt{16} = 48\sqrt{6}\) which is not equal to \(4\sqrt{3}\). Therefore choice D is not equivalent to \(\sqrt{48}\).

Key Concepts

Simplification of Radicals

Radicals often look daunting with their long square roots, but simplification can make them much easier to work with! Imagine you want to find a simpler expression for \(\sqrt{48}\). To do this, identify perfect squares that are factors of 48. One way to start is by breaking down 48 into its prime factors, which are 2 and 3. So, \(48 = 2^4 \times 3\).
From here, look for perfect square factors, which in this case are \(2^4 = 16\). So, rewrite \(\sqrt{48}\) as \(\sqrt{16 \times 3}\).
Now, simplify it further by taking the square root of the perfect square: \(\sqrt{16} = 4\). This gives you \(4\sqrt{3}\).

• This step-by-step simplification shows how radicals can be broken down into easier, more manageable expressions.
• It illustrates the power of recognizing perfect squares within the radical to make the equation simpler.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using mathematical operations. They're like puzzles, requiring you to solve or simplify to understand the value they represent. When examining \(\sqrt{48}\) or any algebraic expression involving radicals, breaking it into simpler forms is key.
Use operations such as multiplication or factorization to see hidden patterns or common factors. In the given options for the exercise, each expression required a breakdown or simplification into a basic understanding of \(\sqrt{48}\):

• Option (A): \(\sqrt{2} \cdot \sqrt{24}\) can further be simplified into \(\sqrt{48}\).
• Option (B): \(2\sqrt{12}\) simplifies to \(4\sqrt{3}\).
• Option (C): Already simplified as \(4\sqrt{3}\).
• Option (D): \(12\sqrt{16}\) improperly attempts to equal \(\sqrt{48}\).

This method demonstrates how rewriting or rearranging algebraic expressions allows you to compare and contrast their equivalences. It also showcases how you don’t always need to solve them, but you can also simplify or transform them.

Mathematical Problem Solving

Problem solving in mathematics is about employing strategies and logical steps to find solutions. When presented with the task of identifying which expression does not equal \(\sqrt{48}\), you need to use logical reasoning. The process starts by simplifying \(\sqrt{48}\), as shown in the previous sections, to \(4\sqrt{3}\).
Next, carefully inspect each of the given options by simplifying or transforming each expression to match \(4\sqrt{3}\). Understanding mathematical equivalency is crucial:

• In Option A and B, breakdown and simplify operations confirm they match \(\sqrt{48}\).
• Option C already straightforwardly presents \(4\sqrt{3}\).
• Option D's complexity indicates a faulty equivalence due to scaling \(48\sqrt{6}\) from \(12\sqrt{16}\) incorrectly.

This thought process helps you identify logical and mathematical steps to each conclusion: a necessary skill in tackling similar algebraic problems. The simplification and logical reasoning are both important tools in your problem-solving toolkit, particularly when dealing with radicals and algebraic expressions.