Problem 131 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I simplified the terms of \(2 \sqrt{20}+4 \sqrt{75},\) and then I was able to add the like radicals.

Step-by-Step Solution

Answer

The statement does not make sense as the simplified terms aren't 'like radicals' and so cannot be further added.

Step 1: Simplify each term

To assess the statement's merit, start by simplifying each term individually. The number under each square root, also known as radicand, can be simplified if it contains perfect square factors. \(2 \sqrt{20}\) becomes \(2*2\sqrt{5} = 4 \sqrt{5}\), since \(20 = 4 * 5\) and the square root of \(4\) is \(2\). Likewise, \(4 \sqrt{75}\) simplifies to \(4 * 5 \sqrt{3} = 20 \sqrt{3}\), since \(75 = 25 * 3\) and the square root of \(25\) is \(5\).

Step 2: Evaluating the statement

According to the student's statement, after simplification 'like radicals' were added. So, it implies that the resulting simplified radicals were similar. However, in step one, it was derived that simplified form of given statement gives \(4 \sqrt{5}\) and \(20 \sqrt{3}\) which clearly aren't 'like radicals.' Like radicals means terms with same radicands, so that they can be added or subtracted like normal algebraic like terms.

Step 3: Final verdict on statement

Based on the above steps, we can say the original statement 'I simplified the terms of \(2 \sqrt{20}+4 \sqrt{75}\), and then I was able to add the like radicals' does not make sense as the simplified forms do not provide 'like radicals.' The terms cannot be further consolidated by adding as they aren't like radicals.

Key Concepts

Like Radicals

When dealing with radical expressions, like radicals are very similar to like terms in algebra. For radicals, like radicals have the same radicand.

The radicand is the number or expression under the square root symbol. Therefore, in order for radicals to be considered 'like,' their radicands must match exactly.

Here are some key points about like radicals:

• Like radicals can be added or subtracted just like like terms in algebra.
• For example, \(3\sqrt{5} + 7\sqrt{5}\) results in \(10\sqrt{5}\) because both terms share the same radicand.
• Improperly identifying like radicals can lead to incorrect simplifications in an expression.

In expressions such as \(4 \sqrt{5}\) and \(20 \sqrt{3}\), since the radicands \(5\) and \(3\) are different, they are not like radicals and cannot be combined by addition or subtraction.

Perfect Square Factors

The ability to simplify radical expressions often depends on identifying perfect square factors of the radicand. A perfect square is a number whose square root is an integer.

The goal is to find the largest perfect square factor of a radicand, as this can simplify the square root expression significantly.

Here's how to identify and use perfect square factors effectively:

• A perfect square factor is a factor that is a square of an integer, such as 4, 9, 16, 25, etc.
• To simplify \(\sqrt{20}\), identify the perfect square factor 4, because \(20 = 4 \times 5\).
• The expression becomes \(\sqrt{4}\sqrt{5} = 2\sqrt{5}\).
• This principle applies to more complex expressions as well, like \(\sqrt{75}\), where 25 is the perfect square factor, simplifying to \(5\sqrt{3}\).

By extracting the perfect square factor, you simplify the expression, making it easier to handle further calculations or to identify like radicals.

Simplifying Square Roots

Simplifying square roots is a foundational operation in algebra that transforms complex radical expressions into simpler forms. This makes calculations easier and solutions more precise.

Here's a step-by-step approach to simplify square roots effectively:

• Look for the largest perfect square factor of the radicand.
• Rewrite the radicand as a product of this perfect square and another factor.
• Take the square root of the perfect square factor out of the square root sign.

For example, consider simplifying \(\sqrt{75}\). You can express 75 as \(25 \times 3\).

Then \(\sqrt{75}\) becomes \(\sqrt{25} \times \sqrt{3} = 5\sqrt{3}\).

This process not only simplifies the expression but makes it much straightforward to work with in further operations or comparisons, as seen when dealing with radical addition or subtraction.