Problem 33 - 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. $$ \sqrt{8 a}-\sqrt{2 a} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sqrt{2a} \).

Step 1: Simplify the Radicals

First, simplify each square root separately. For \( \sqrt{8a} \), we can break it into \( \sqrt{4} \times \sqrt{2a} \). Since \( \sqrt{4} = 2 \), it simplifies to \( 2\sqrt{2a} \). The expression \( \sqrt{2a} \) remains the same since it cannot be simplified further.

Step 2: Substitute and Combine Like Terms

Now replace \( \sqrt{8a} \) with \( 2\sqrt{2a} \) in the expression: \( 2\sqrt{2a} - \sqrt{2a} \). Since both terms are like terms, you can subtract: \((2-1)\sqrt{2a} = \sqrt{2a} \).

Key Concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, return the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. Understanding square roots is essential when simplifying radical expressions.

• A square root is often indicated by the radical symbol (\(\sqrt{}\)).
• The number inside the radical sign is known as the radicand.
• Square roots of perfect squares such as 4, 9, and 16 are whole numbers, making them easy to simplify.

Knowing about square roots helps in breaking down and simplifying larger expressions effectively.

Radicands

A radicand is the number or expression inside a radical symbol (\(\sqrt{}\)). It is the part of the radical expression that we want to find the root of.

• The radicand can be a whole number, a variable, or a combination of both.
• In the context of square roots, simplifying the radicand can often simplify the entire expression.

For instance, in the expression \(\sqrt{8a}\), 8a is the radicand.To simplify, you might look for factors of the radicand that are perfect squares, as seen in \(\sqrt{8a} = \sqrt{4} \times \sqrt{2a}\), where 4 is a perfect square.

Like Terms

Like terms are terms in an expression that have the same variables raised to the same power, allowing them to be combined.In the context of radicals, like terms have the same radicand.For example, in the expression \(2\sqrt{2a} - \sqrt{2a}\), both terms are like terms because they share the same radicand, \(2a\).

• To combine like terms, simply add or subtract their coefficients.
• Only like terms can be combined to simplify expressions.

By recognizing like terms within an expression, you can effectively streamline and simplify your work.

Expressions

Expressions in mathematics are a combination of numbers, variables, and operators (like addition and subtraction).Expressions can range from simple to complex and often include numbers and variables raised to powers or within radicals.

• In expressions like \(\sqrt{8a} - \sqrt{2a}\), terms are simplified and combined.
• The goal is to rewrite expressions in their simplest and most recognizable form.

By focusing on simplifying each part of the expression, you make the problem-solving process easier and clearer.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in a simpler or more efficient form.This involves breaking down complex parts and combining like terms. For our given expression \(\sqrt{8a} - \sqrt{2a}\), the process involves:1. Breaking down and simplifying square roots when possible.2. Replacing any simplifiable radicals.3. Combining like terms to reduce the expression to its simplest form.

• Simplifying makes expressions easier to understand and work with.
• It reduces errors in calculations by breaking them into manageable parts.

Knowing how to simplify expressions will enhance your algebra skills and improve your ability to solve mathematical problems efficiently.