Simplify The Expression: \[ \frac{2 X^0 \times (2 X)^0}{2(x Y Z)^3} \]

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex-looking mathematical statements and reduce them to their most basic and understandable form. This process often involves applying various rules and properties of arithmetic and algebra. One such property that often comes into play is the zero exponent rule, which states that any non-zero number raised to the power of zero is equal to 1. Additionally, we need to understand how exponents interact with products and quotients of variables. This article delves into the simplification of an algebraic expression involving zero exponents, variables, and the power of a product rule. Let's break down the expression and simplify it step-by-step, highlighting the key concepts and techniques involved.

The given expression is: ${ \frac{2 x^0 \times (2 x)^0}{2(x y z)^3} }$

This expression involves several components: constants, variables raised to powers, and products of variables. Our goal is to simplify this expression by applying the rules of exponents and basic algebraic manipulations. We will begin by addressing the zero exponents, then move on to simplifying the power of a product, and finally, we will combine like terms to arrive at the simplest form of the expression. Understanding the order of operations and the properties of exponents is crucial for accurate simplification. This article will provide a clear and concise explanation of each step, making the process accessible to anyone with a basic understanding of algebra. Through this example, we aim to reinforce the importance of simplification in mathematics and demonstrate how it can reveal the underlying simplicity of complex expressions.

Step-by-Step Simplification

1. Applying the Zero Exponent Rule

The zero exponent rule is a cornerstone of simplifying expressions. It states that any non-zero number raised to the power of zero is equal to 1. Mathematically, this is represented as a⁰ = 1, where a ≠ 0. This rule dramatically simplifies expressions because it allows us to replace terms with exponents of zero with the number 1. In our given expression, we have two terms with zero exponents: x⁰ and (2x)⁰. Applying the zero exponent rule to these terms, we get:

• x⁰ = 1 (assuming x ≠ 0)
• (2x)⁰ = 1 (assuming x ≠ 0)

Substituting these values back into the original expression, we get: ${ \frac{2 \times 1 \times 1}{2(x y z)^3} }$

This substitution simplifies the numerator significantly, reducing it from a product involving exponents to a simple multiplication of constants. The expression now looks much cleaner and easier to work with. The next step involves simplifying the denominator, which contains a power of a product. Before proceeding, it is important to acknowledge the assumption that x ≠ 0. The zero exponent rule is only valid for non-zero bases. This is a crucial detail to keep in mind when simplifying expressions, as overlooking such conditions can lead to incorrect results. In the context of problem-solving, stating these assumptions explicitly demonstrates a thorough understanding of the underlying mathematical principles.

2. Simplifying the Numerator

After applying the zero exponent rule, the numerator of our expression has become a simple product of constants: 2 × 1 × 1. This multiplication is straightforward and results in the number 2. Therefore, the numerator simplifies to 2. The expression now looks like this: ${ \frac{2}{2(x y z)^3} }$

This simplification highlights the power of the zero exponent rule in reducing the complexity of expressions. By replacing the terms with zero exponents with 1, we have effectively eliminated those terms from the numerator, making the expression more manageable. The next step involves simplifying the denominator, which contains a product of variables raised to a power. The denominator presents a slightly more complex challenge, as we need to apply the power of a product rule before we can further simplify the expression. However, with the numerator now in its simplest form, we can focus our attention on the denominator and systematically apply the relevant rules to achieve further simplification. The ultimate goal is to combine like terms and reduce the expression to its most concise form, which will reveal the underlying relationship between the variables and constants involved.

3. Applying the Power of a Product Rule in the Denominator

The denominator of our expression is 2(x y z)³. To simplify this, we need to apply the power of a product rule. This rule states that when a product of terms is raised to a power, each term within the product is raised to that power. Mathematically, this is represented as (a b)ⁿ = aⁿ bⁿ. Applying this rule to our denominator, we get:

( x y z )³ = x³ y³ z³

Substituting this back into the denominator, the expression becomes:

2(x y z)³ = 2x³y³z³

Now, the entire expression looks like this: ${ \frac{2}{2 x^3 y^3 z^3} }$

This step is crucial because it distributes the exponent to each variable within the parentheses, allowing us to further simplify the expression. The power of a product rule is a fundamental tool in algebraic manipulation, and its correct application is essential for achieving accurate results. By expanding the denominator in this way, we have set the stage for the final step, which involves canceling common factors between the numerator and the denominator. This final simplification will reveal the most concise form of the expression and provide a clear understanding of the relationships between the variables involved. The careful and methodical application of the power of a product rule demonstrates a solid grasp of algebraic principles.

4. Canceling Common Factors

Now we have the expression: ${ \frac{2}{2 x^3 y^3 z^3} }$

We can see that there is a common factor of 2 in both the numerator and the denominator. We can cancel this common factor by dividing both the numerator and the denominator by 2: ${ \frac{2 \div 2}{2 x^3 y^3 z^3 \div 2} = \frac{1}{x^3 y^3 z^3} }$

This step significantly simplifies the expression by eliminating the constant factor. Canceling common factors is a fundamental technique in simplifying fractions and algebraic expressions. It allows us to reduce the expression to its simplest form, making it easier to understand and work with. In this case, canceling the factor of 2 reveals the core relationship between the variables in the expression. The expression is now in its most simplified form, with no more common factors to cancel. This final form clearly shows the inverse relationship between the constant 1 and the product of the variables raised to the power of 3. The ability to identify and cancel common factors is a key skill in algebraic manipulation and is essential for solving a wide range of mathematical problems.

Final Simplified Expression

After performing all the simplification steps, we have arrived at the final simplified expression: ${ \frac{1}{x^3 y^3 z^3} }$

This is the simplest form of the original expression. It clearly shows the relationship between the variables x, y, and z. The expression represents the reciprocal of the product of these variables, each raised to the power of 3. This simplified form is much easier to understand and work with compared to the original expression. The process of simplification has revealed the underlying structure of the expression, making it more accessible and less intimidating. Understanding how to simplify expressions is crucial in mathematics, as it allows us to solve complex problems more easily and gain deeper insights into mathematical relationships. This example demonstrates the power of applying basic algebraic rules and properties to reduce complex expressions to their most fundamental forms. The final simplified expression serves as a clear and concise representation of the original mathematical statement.

Conclusion

In conclusion, we successfully simplified the given expression: ${ \frac{2 x^0 \times (2 x)^0}{2(x y z)^3} }$

by applying the zero exponent rule, the power of a product rule, and canceling common factors. The final simplified expression is: ${ \frac{1}{x^3 y^3 z^3} }$

This process highlights the importance of understanding and applying the fundamental rules of algebra. Simplification is a key skill in mathematics, enabling us to reduce complex expressions to their simplest forms, making them easier to understand and work with. By breaking down the problem into smaller steps and applying the appropriate rules, we can systematically simplify even the most challenging expressions. The zero exponent rule, which states that any non-zero number raised to the power of zero is equal to 1, played a crucial role in reducing the complexity of the numerator. The power of a product rule allowed us to distribute the exponent in the denominator, further simplifying the expression. Finally, canceling common factors between the numerator and the denominator resulted in the most concise form of the expression. This example serves as a valuable illustration of how algebraic simplification can reveal the underlying structure of mathematical statements and make them more accessible. Mastering these simplification techniques is essential for success in mathematics and related fields.