Problem 7 - 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. $$ \sqrt{50}+\sqrt{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(6\sqrt{2}\).

Step 1: Factor the Radicand

First, we need to simplify \(\sqrt{50}\). To do this, identify the factors of 50. We can express 50 as \(25 \times 2\), where 25 is a perfect square.

Step 2: Simplify the Square Root

Next, take the square root of the perfect square 25. Since \(\sqrt{25} = 5\), we can simplify \(\sqrt{50}\) as follows: \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\).

Step 3: Combine Like Terms

Now, we need to combine the simplified form \(5\sqrt{2}\) with the original \(\sqrt{2}\) in the expression \(5\sqrt{2} + \sqrt{2}\). This can be written as \(5\sqrt{2} + 1\sqrt{2} = 6\sqrt{2}\).

Key Concepts

Square Roots

Square roots are mathematical expressions that help us find a number which, when multiplied by itself, gives the original number. The square root is represented by a radical symbol \(\sqrt{\ }\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

• A square root indicates a number that produces a specified quantity when squared.
• Simplifying square roots often involves finding perfect square factors.
• In expressions, simplifying square roots makes calculations easier.

Recognizing perfect square factors quickly simplifies square root expressions, minimizing complex calculations.

Like Terms

Like terms are terms that have the same variables raised to the same power. They can be combined through addition or subtraction because they represent the same quantity but perhaps with different coefficients. This principle is key in simplifying expressions like \(5\sqrt{2} + \sqrt{2}\).

• Terms with exactly the same variable part can be simplified or "combined".
• In expressions involving square roots, terms like \(5\sqrt{2}\) and \(\sqrt{2}\) are considered like terms because they both contain \(\sqrt{2}\).
• Combining like terms simplifies expressions to their most reduced form for easy comprehension and further calculation.

In practice, if you identify like terms, you streamline the expression drastically by consolidating coefficients.

Radicand

The radicand is the number inside the radical sign that we are interested in taking the root of. For example, in \(\sqrt{50}\), the radicand is 50. Understanding the components of the radicand is integral to operations such as simplifying square roots.

• Knowing how to identify a radicand involves observing which number is under the square root.
• Simplifying the radicand may involve breaking it down into its perfect square factors.
• By factoring the radicand, you make square root simplification straightforward.

Focusing on the radicand first allows for targeted simplification approaches that can simplify complex expressions.

Perfect Squares

Perfect squares are numbers that are squares of integers. For instance, \(4\), \(9\), \(16\), \(25\), and so on. Recognizing perfect squares is crucial when simplifying square roots, as they reduce calculations significantly.

• In simplification, identify perfect squares within a radicand for easy square root extraction.
• Finding perfect squares is the golden strategy in breaking down complex roots.
• As seen in \(\sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\), knowing \(25\) is a perfect square simplifies the root.

Recognizing perfect squares allows you to easily transform and simplify numerical expressions, making the overall computation process more efficient.