Problem 17 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 4 \sqrt{5}+\sqrt{3}+\sqrt{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(5 \sqrt{5} + \sqrt{3}\).

Step 1: Identify like radicals

Look for radicals that have the same radicand. In this case, \(4 \sqrt{5}\) and \(\sqrt{5}\) are like radicals, because they both have '5' as the radicand.

Step 2: Combine like radicals

The like radicals can be combined by adding or subtracting the coefficients. Here, we add the coefficients of the \(\sqrt{5}\) terms: \(4 \sqrt{5} + \sqrt{5} = 5 \sqrt{5}\).

Step 3: Write the final simplified expression

Now, rewrite the expression using the newly combined like terms, leaving any other terms as they are. In this case, the final expression is: \(5 \sqrt{5} + \sqrt{3}\).

Key Concepts

Understanding Radicals

Radicals are an essential part of mathematics, often represented with a square root symbol, such as \( \sqrt{} \). The term under the square root is known as the radicand. For example, in \( \sqrt{5} \), the radicand is 5. Radicals are unique because they represent roots of numbers, which is a fundamental concept in algebra.

• Radicals help in expressing numbers that aren't perfect squares in a simplified form.
• Square roots are the most common radicals, but they can extend to cubic roots and beyond.

Understanding how radicals function is crucial in algebra. They help in simplifying large expressions and solving equations. When dealing with radicals, always aim to simplify by finding square factors within the radicand or by combining like terms as seen in expressions like \( \sqrt{5} \).
Radicals can be simplified if the radicand has square factors. For instance, \( \sqrt{20} \) can be broken down to \( \sqrt{4 \times 5} = 2\sqrt{5} \). Learning to recognize these factors makes handling complex expressions easier.

Working with Coefficients

Coefficients in mathematics are numbers that multiply a variable or radical. In an expression like \( 4 \sqrt{5} \), 4 is the coefficient. Coefficients provide weight or value to the term they are attached to.

• They are essential in operations like addition and subtraction of terms.
• They determine how many times the radical or variable should be counted.

When simplifying expressions involving radicals, coefficients can be added or subtracted when their radicals are the same, as they act like multipliers for the radical part. In the example \( 4 \sqrt{5} + \sqrt{5} \), you can add the coefficients 4 and 1 since they share the same radical, resulting in \( 5 \sqrt{5} \).
Proper handling of coefficients is key in simplification processes, ensuring that the expressions are reduced to their simplest forms efficiently. Coefficients also play a significant role in evaluating expressions in algebraic equations and inequalities.

The Concept of Like Terms

In algebra, like terms refer to terms that have the same variable part, with the same exponents. When it comes to radicals, like terms have the same radicand. This means their root values are identical, allowing them to be combined through addition or subtraction. For example, both \( 4\sqrt{5} \) and \( \sqrt{5} \) have the same radicand of 5, making them like terms.

• Identifying like terms allows for the simplification of expressions.
• It helps in minimizing complex algebraic equations and solving them with ease.

Adding or subtracting like terms is a straightforward process. You simply deal with the coefficients while the radical part remains unchanged. For instance, \( 4 \sqrt{5} + \sqrt{5} = 5\sqrt{5} \). This is similar to how you might add \( 4x + x = 5x \) in variable terms.
Recognizing and combining like terms in expressions is a fundamental algebraic skill. It prevents errors and simplifies problem-solving, allowing students to handle more complex algebraic challenges successfully.