Problem 48 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{4 x^{2} y^{2} z^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given radical expression: $$\sqrt{4x^2y^2z^2}$$. Answer: The simplified radical expression is $$2xyz$$.

Step 1: Identify the powers of the variables

The given expression is $$\sqrt{4x^2y^2z^2}$$. Notice that the powers of each variable (x, y, and z) are all even numbers.

Step 2: Apply the rule for square roots of products

According to the rule for square roots of products, we can apply the square root individually to each number and variable. The given expression then can be simplified as: $$ \sqrt{4x^2y^2z^2} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^2} \cdot \sqrt{z^2} $$

Step 3: Simplify the square roots

Now, simplify each term as follows: $$ \sqrt{4} = 2 \\ \sqrt{x^2} = x \\ \sqrt{y^2} = y \\ \sqrt{z^2} = z $$

Step 4: Combine the terms

Finally, combine the terms to obtain the simplified expression: $$ 2 \cdot x \cdot y \cdot z = 2xyz $$ The simplified radical expression is $$2xyz$$.

Key Concepts

Square Roots of Products

Understanding how to simplify the square roots of products is a fundamental skill in algebra. When we have a square root covering a product such as \( \sqrt{ab} \) where \( a \) and \( b \) are both positive numbers or variables, the rule is that you can take the square root of each factor separately. In other words, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \). This is a direct application of the product rule for radicals. It's important to remember that this rule only applies when all the quantities under the square root are non-negative, as square roots of negative numbers introduce complex numbers, which are beyond basic algebra.

For instance, in our example \( \sqrt{4 x^{2} y^{2} z^{2}} \), this rule lets us break down the radical into simpler parts: \( \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^2} \cdot \sqrt{z^2} \). By doing this, we can focus on simplifying each square root one at a time, which considerably simplifies the process and avoids confusion.

Even Powers in Algebra

Handling even powers in algebra, such as \( x^2, y^4, z^6 \), etc., is much simpler than dealing with odd powers or non-integer exponents because when you take the square root of an even power, the result is always the base number with half the exponent. This is because the square root function is the inverse of squaring a number.

For example, \( \sqrt{x^2} = x \) and \( \sqrt{x^4} = x^2 \). It is crucial to ensure the variables represent non-negative values, as negative numbers raised to an even power can complicate the square root due to their positive results. Moreover, it signifies that you can directly remove the square root and reduce the exponent by half when dealing with even powers under a square root, which is what makes simplifying radical expressions involving even powers so straightforward.

Radical Simplification Process

The radical simplification process involves several steps, each focused on methodically breaking down and simplifying expressions under the radical sign. The process usually begins with identifying the components inside the radical that can be simplified—such as perfect squares or even powers of variables—and then applying the appropriate rules for simplification.

Following a structured process helps ensure that no mistakes are made and the simplest form is reached. The steps in our example problem involved identifying the powers of each variable, applying the square roots of products rule, simplifying each square root individually, and combining the results.

It can be very helpful to remember that during the simplification process, it's often easier to handle numbers and variables separately. Take care of the numerical coefficients first, as they can be simplified directly with arithmetic. Then, apply the rules to the variables, and after all the parts are simplified, you can multiply them back together to obtain the final simplified expression.