Problem 12 - Intermediate Algebra

Question

For Problems \(1-20\), use the distributive property to help simplify each of the following. For example, $$ \begin{aligned} 3 \sqrt{8}-\sqrt{32} &=3 \sqrt{4} \sqrt{2}-\sqrt{16} \sqrt{2} \\ &=3(2) \sqrt{2}-4 \sqrt{2} \\ &=6 \sqrt{2}-4 \sqrt{2} \\ &=(6-4) \sqrt{2}=2 \sqrt{2} \end{aligned} $$ $$ \frac{3}{5} \sqrt{5}-\frac{1}{4} \sqrt{80} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{5} \sqrt{5}\).

Step 1: Break Down Square Roots into Factors

Rewrite the square roots in terms of their prime factors where possible. Start by factoring 80 as follows: \(80 = 2^4 \times 5\). This gives us the equivalent form for the square roots: \(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\).

Step 2: Simplify Each Term

Substitute the factored square root back into the expression: \(\frac{3}{5} \sqrt{5} - \frac{1}{4}(4 \sqrt{5})\). Simplifying gives \(\frac{3}{5} \sqrt{5} - \sqrt{5}\).

Step 3: Apply the Distributive Property

Factor out the common term \(\sqrt{5}\) from the expression: \(\left(\frac{3}{5} - 1\right) \sqrt{5}\). Simplifying the terms within the parentheses results in \(\left(\frac{3}{5} - \frac{5}{5}\right) \sqrt{5}\).

Step 4: Perform the Subtraction in the Parentheses

Subtract the fractions: \(\frac{3}{5} - \frac{5}{5} = -\frac{2}{5}\). Now our expression becomes \(-\frac{2}{5} \sqrt{5}\).

Key Concepts

Simplifying Expressions

Simplifying expressions, especially those involving square roots or fractional coefficients, can seem complex. However, there are systematic ways to break these down into simpler forms. The aim is to make expressions easier to understand or use. Here's how we can simplify:

• Identify and Separate Terms: First, determine each term in the expression. In our example, the terms are \(\frac{3}{5} \sqrt{5}\) and \(-\frac{1}{4} \sqrt{80}\).


• Rewrite Complex Terms: For terms like \(\sqrt{80}\), break them down using prime factors. This is done to expose any perfect squares, making it easier to simplify further. For instance, \(80\) can be written in terms of prime factors as \(2^4 \times 5\).


• Factor and Simplify: Once you identify the square component, such as \(\sqrt{16}\) from \(80\), you can simplify the expression to \(4 \sqrt{5}\). Substitute back into the original expression to replace complex terms with their simplified forms.

Following these steps can help break down even the most daunting of expressions into something more manageable.

Factoring Square Roots

Factoring square roots involves expressing a number as a product of its largest possible square factor and another number. This technique is essential when simplifying expressions involving square roots.

• Identify Prime Factors: Start by identifying the prime factors of the number under the square root. In our example, we factor \(80\) as \(2^4 \times 5\).


• Look for Square Components: Identify any perfect squares from the prime factors. Here, \(2^4\) gives us \(16\), which has a square root of \(4\).


• Rewrite Using Square Roots: Express the original square root as the product of two simpler square roots: \(\sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\). This separation allows for straightforward simplification.

Through this process, you can tackle any square root and simplify it for easier calculations or further algebraic operations.

Algebraic Operations

Algebraic operations in simplifying expressions often involve the use of the distributive property and working with fraction manipulation effectively. Understanding these operations is crucial for successfully simplifying expressions.

• The Distributive Property: This principle allows us to factor out common terms. In our exercise, \(\sqrt{5}\) is a common term in both parts of the expression, allowing us to factor it out efficiently.


• Handling Fractions: Operations with fractions, such as addition or subtraction, often require a common denominator. In our example, subtract \(\frac{5}{5}\) from \(\frac{3}{5}\), which gives a simple fraction subtraction: \(-\frac{2}{5}\).


• Putting It All Together: By combining like terms (e.g., common factors like \(\sqrt{5}\)) and simplifying within parentheses, you reduce expressions to their simplest form. This transformation makes it easier to interpret and use the expression in further calculations.

Mastering these operations allows one to handle more complex algebraic expressions confidently, ensuring the simplest form is achieved.