Problem 6 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 4 \sqrt{32}-\sqrt{18}+2 \sqrt{128} $$

Step-by-Step Solution

Answer

The simplified expression is \( 29\sqrt{2} \).

Step 1: Simplify the Square Roots

First, simplify the square roots in the expression: \[4 \sqrt{32}, \quad -\sqrt{18}, \quad 2 \sqrt{128}.\]To simplify \( \sqrt{32} \), note that \( 32 = 16 \times 2 \) and \( \sqrt{16} = 4 \). Thus, \[ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}. \]To simplify \( \sqrt{18} \), note that \( 18 = 9 \times 2 \) and \( \sqrt{9} = 3 \). Thus, \[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}. \]To simplify \( \sqrt{128} \), note that \( 128 = 64 \times 2 \) and \( \sqrt{64} = 8 \). Thus, \[ \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}. \]Now the expression becomes:\[4(4\sqrt{2}) - 3\sqrt{2} + 2(8\sqrt{2}).\]

Step 2: Distribute and Combine Terms

Next, distribute the constants to each square root in the expression:\[4 \times 4\sqrt{2} = 16\sqrt{2},\]\[3\sqrt{2} = 3\sqrt{2},\]\[2 \times 8\sqrt{2} = 16\sqrt{2}.\]The expression becomes:\[16\sqrt{2} - 3\sqrt{2} + 16\sqrt{2}.\]

Step 3: Combine Like Terms

Now, combine the like terms. These are terms that have the same radical component (\(\sqrt{2}\)):\[(16\sqrt{2} + 16\sqrt{2} - 3\sqrt{2}) = (16 + 16 - 3)\sqrt{2} = 29\sqrt{2}.\]

Step 4: Final Answer

The final simplified expression is:\[ 29\sqrt{2}. \]

Key Concepts

Simplifying Square Roots

Simplifying square roots can initially seem a bit tricky, but once you understand the process, it becomes much easier. The goal is to express a square root in its simplest form. This is done by breaking down the number inside the square root into its prime factors. Then identify perfect squares among these factors.
For instance, consider simplifying \( \sqrt{32} \). Note that \(32 = 16 \times 2\), and since \(16\) is a perfect square (as \(4 \times 4 = 16\)), \( \sqrt{16} = 4\). Thus, you can simplify \( \sqrt{32} \) to \( 4\sqrt{2} \). This process is done similarly for \( \sqrt{18} \) and \( \sqrt{128} \) by finding and extracting the largest perfect square factors from inside the square roots.

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as square roots or cube roots. In algebra, these expressions often involve numbers, variables, or both under the radical sign, making them more complex.
In the given exercise, we simplified radical expressions like \( \sqrt{32} \), \( \sqrt{18} \), and \( \sqrt{128} \). Breaking these down into simpler terms like \( 4\sqrt{2} \), \( 3\sqrt{2} \), and \( 8\sqrt{2} \) helps us combine and manipulate them more easily. Once simplified, these radical expressions often involve similar terms, making the process of combination straightforward.

Real Numbers

Real numbers encompass a broad set of numbers that include both rational numbers (like fractions and integers) and irrational numbers (which cannot be expressed as exact fractions, such as \( \sqrt{2} \) and \( \pi \)).
In dealing with radical expressions, we typically assume all variables represent positive real numbers to avoid issues with undefined expressions or complex numbers. For example, in our exercise, assumptions about the positivity of real numbers simplify the computations, ensuring \( \sqrt{16} \) results in \(4\) rather than needing to consider negative roots or imaginary numbers.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. This involves adding or subtracting terms that have the same variables and powers. In radical expressions, like terms share the same radical part.
For example, in the expression \(16\sqrt{2} - 3\sqrt{2} + 16\sqrt{2}\), all terms are like terms because they each have \(\sqrt{2}\) as their radical. You can simply add the coefficients: \((16 - 3 + 16)\sqrt{2} = 29\sqrt{2}\). This process significantly reduces the complexity of an expression, offering a clear and concise final result.