Problem 69 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 3 \sqrt{28}+3 \sqrt{7} $$

Step-by-Step Solution

Answer

The simplified form is \( 9\sqrt{7} \).

Step 1: Factor the Radicals

Begin by factoring the radicand (the number inside the square root) in the expression. For \( 28 \) inside \( \sqrt{28} \), we break it down into \( 28 = 4 \times 7 \). Since \( 4 \) is a perfect square, we can rewrite \( \sqrt{28} \) as \( \sqrt{4 \times 7} \).

Step 2: Simplify the Radical Expressions

Using the property \( \sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b} \), simplify \( \sqrt{28} \) into \( \sqrt{4} \times \sqrt{7} \). We know \( \sqrt{4} = 2 \), so \( \sqrt{28} = 2\sqrt{7} \).

Step 3: Replace and Combine Like Terms

Substitute \( 2\sqrt{7} \) back into the expression: \[ 3 \sqrt{28} + 3 \sqrt{7} = 3(2\sqrt{7}) + 3\sqrt{7} \].Now simplify by distributing:\[ 6\sqrt{7} + 3\sqrt{7} \].

Step 4: Add Like Terms

Combine the like terms. Since both terms have \( \sqrt{7} \), add the coefficients:\[ 6\sqrt{7} + 3\sqrt{7} = (6 + 3)\sqrt{7} \]. Thus, the expression becomes:\[ 9\sqrt{7} \].

Key Concepts

Perfect Square Factorization

When simplifying radicals, looking for perfect square factors is key. A perfect square is any number that is the square of an integer, like 4, 9, or 16. To simplify a radical expression, you begin by factoring the radicand. This means you break down the number inside the square root into factors.

• For example, in the radical \( \sqrt{28} \), 28 can be factored into \( 4 \times 7 \).
• Since 4 is a perfect square (because \( 2^2 = 4 \)), we can pull it out of the radical.

The expression \( \sqrt{4 \times 7} \) can then be rewritten using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). By doing this, the expression becomes \( \sqrt{4} \times \sqrt{7} \), which simplifies further to \( 2\sqrt{7} \) because \( \sqrt{4} = 2 \). This process reduces the complexity of the radical, making it easier to solve.

Combining Like Terms

Combining like terms is a fundamental aspect when simplifying algebraic expressions. It involves combining terms that have the same variables and powers. In the context of radicals:

• Only terms with the same radical can be combined.
• This maximizes the simplicity of the expression.

For example, in the expression \( 3\sqrt{28} + 3\sqrt{7} \), once you've simplified \( \sqrt{28} \) to \( 2\sqrt{7} \), your expression becomes \( 3(2\sqrt{7}) + 3\sqrt{7} \). Now, distribute the 3 to get \( 6\sqrt{7} + 3\sqrt{7} \). Both terms contain \( \sqrt{7} \), making them "like terms." Therefore, you can add their coefficients: \( 6 + 3 \), giving you \( 9\sqrt{7} \). This consolidation into one term simplifies the expression further.

Properties of Square Roots

To efficiently simplify expressions involving square roots, one must grasp their properties. These properties aid both in breaking down and combining expressions.

• The property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) allows you to separate components within a square root, simplifying computations.
• Similarly, \( \sqrt{a^2} = a \) emphasizes that the square and square root are inverse operations. This is particularly useful when working with perfect squares, as shown in previous steps.

There's also the property that allows for combining terms with identical radicands. For instance, only terms like \( x\sqrt{y} + z\sqrt{y} \) can be combined because they share the same radical component \( \sqrt{y} \). By understanding and utilizing these properties, you can manipulate and reduce radical expressions to their simplest forms, just as in the original exercise, where \( 3\sqrt{28} + 3\sqrt{7} \) was simplified to \( 9\sqrt{7} \).