Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 5 \sqrt{7}-\sqrt{7} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(4\sqrt{7}\).

Step 1: Identify Common Terms

Notice that both terms in the expression \(5\sqrt{7}-\sqrt{7}\) have the same radical, \(\sqrt{7}\). This means they can be combined, just as with like terms in algebra.

Step 2: Apply Arithmetic on Coefficients

Since the radicals are the same, we can subtract the coefficients. The expression changes to \((5-1)\sqrt{7}\).

Step 3: Simplify Coefficients

Subtract the coefficients: \(5 - 1 = 4\). Thus, the expression simplifies to \(4\sqrt{7}\).

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, combining like terms is a crucial step in simplifying. In the exercise provided, the terms were given as \(5\sqrt{7}-\sqrt{7}\). These terms are simplified by identifying terms that share the same properties, particularly having the same radical part. Here, the radical \(\sqrt{7}\) is common in both terms, which classifies them as 'like terms'.

• Like Terms: These are terms with the same variable part, in this case, the same radical expression \(\sqrt{7}\).
• Combining: To combine, simply add or subtract the coefficients. In this instance, we subtract the coefficients since the problem uses a subtraction operation.

By identifying and combining like terms, we can efficiently simplify expressions while handling fewer terms. This process reduces the complexity of algebraic expressions and helps in keeping calculations straightforward.

Radical Expressions

Radical expressions include symbols such as roots—square roots, cube roots, etc. In our problem, we are working with a square root: \(\sqrt{7}\). Simplifying radical expressions often involves reducing them to their simplest form. Here's a closer look at what this entails:

• Radical Notation: The square root of a number \(x\), noted as \(\sqrt{x}\), represents a value that, when multiplied by itself, gives \(x\).
• Radicand: The number inside the radical symbol, in our case, 7. In this problem, we can't simplify \(\sqrt{7}\) into a simpler radical without decimals because 7 is prime.

Identifying radicals that can be combined or further simplified helps us in various algebraic operations. The emphasis here is on maintaining the same radicand when aiming to simplify by combining coefficients.

Arithmetic Operations

At the core of simplifying expressions like \(5\sqrt{7}-\sqrt{7}\) is performing basic arithmetic operations. These operations, such as addition, subtraction, multiplication, and division, directly affect the coefficients in such expressions.Steps in Arithmetic Operations with Coefficients:

• Identification: Recognize similar radicals.
• Operation: Apply the required arithmetic operation on the coefficients. Here, it's subtraction: \(5 - 1\).
• Result: Compute the result, yielding 4, resulting in the expression \(4\sqrt{7}\).

Applying these elementary arithmetic principles allows for straightforward manipulation of terms, ensuring expressions are simplified efficiently. Keeping arithmetic operations precise ensures clarity in problem-solving.