Problem 30 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{x} \sqrt{x} $$

Step-by-Step Solution

Answer

Answer: The product of the expression \(\sqrt{x}\sqrt{x}\) is \(x\).

Step 1: Identify the terms to be multiplied

We are given the expression \(\sqrt{x}\sqrt{x}\). Here, the terms to be multiplied are \(\sqrt{x}\) and \(\sqrt{x}\).

Step 2: Multiply the square roots

To multiply the two square roots, we write it like this: $$ \sqrt{x} \cdot \sqrt{x} $$

Step 3: Apply the property of square roots

We can now apply the property of square roots, which states that \(\sqrt{a} \cdot \sqrt{a} = a\). In this case, our "a" is "x", so we have: $$ \sqrt{x} \cdot \sqrt{x} = x $$

Step 4: State the final answer

By multiplying the two square roots together, we have found the product to be: $$ \sqrt{x} \cdot \sqrt{x} = x $$

Key Concepts

Multiplication of Square Roots

When you have square roots being multiplied together, like \(\sqrt{x}\) and \(\sqrt{x}\), it's a straightforward process. Multiplying square roots involves using a property that simplifies multiplication under the radical. If you multiply \(\sqrt{a}\) by \(\sqrt{b}\), you multiply the numbers under the radicals first. This is expressed as:

• \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)
• In simpler cases, such as \(\sqrt{x} \cdot \sqrt{x}\), the solution turns into \(\sqrt{x^2}\).

Keep in mind, \(\sqrt{x} \cdot \sqrt{x}\) simplifies fully into just \(x\) because \(\sqrt{x^2} = x\). This understanding of multiplication makes it easier to manage square roots in algebraic expressions.
Practicing these concepts can build confidence in solving more complex equations involving radicals.

Properties of Square Roots

Square roots have special mathematical properties that simplify how we work with them. One of the main properties helpful in today's exercise is how multiplying square roots together often results in removing the radical. Let's break this property down:

• \((\sqrt{a})^2 = a\): This tells us that whenever you multiply a square root by itself, you get the number under the square root.
• Importantly, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) allows combining square roots during multiplication.

This comes in handy when simplifying expressions, as seen with \(\sqrt{x} \cdot \sqrt{x}\) simply equaling \(x\).
Recognizing these properties allows you to efficiently handle and work through problems involving square roots. Comprehending them can be a game-changer when simplifying expressions.

Basic Algebra

Basic algebra often serves as the stepping stone for understanding more complex mathematical problems, including dealing with square roots. Algebra is concerned with symbols and the rules for manipulating these symbols.

• In algebra, terms like variables are used instead of known numbers. For example, "x" is a common variable.
• When dealing with expressions like \(\sqrt{x} \cdot \sqrt{x}\), the algebraic rules we apply come directly from understanding mathematical properties, like those of square roots.

Performing operations such as multiplication on algebraic terms follows certain principles, as you apply properties rigorously across different problems.
Understanding algebra basics can empower you to simplify and solve expressions smoothly. It is foundational knowledge necessary for dealing with expressions that involve radical terms.