Simplifying Expressions With Order Of Operations

In the realm of mathematics, simplifying expressions is a fundamental skill. One crucial aspect of this simplification process is understanding and applying the order of operations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this order ensures consistency and accuracy in mathematical calculations. This comprehensive guide delves into the intricacies of using the order of operations to simplify expressions, particularly focusing on expressions involving radicals and exponents. We will dissect the expression √((63x^7)/(7x3)), providing a step-by-step walkthrough to illustrate the application of PEMDAS in a practical context.

Understanding the Order of Operations (PEMDAS)

Before we tackle the specific expression, let's solidify our understanding of the order of operations, or PEMDAS. This mnemonic provides a roadmap for simplifying mathematical expressions:

• Parentheses: Operations within parentheses (or other grouping symbols like brackets) are performed first.
• Exponents: Next, we address exponents and radicals.
• Multiplication and Division: These operations are performed from left to right.
• Addition and Subtraction: Finally, addition and subtraction are carried out from left to right.

Mastering PEMDAS is paramount to achieving accurate results in mathematical simplification. By adhering to this order, we eliminate ambiguity and ensure that expressions are evaluated consistently, regardless of who is performing the calculation. Now, let's apply this knowledge to simplify the given expression.

Deconstructing the Expression: √((63x^7)/(7x3))

Our objective is to simplify the expression √((63x^7)/(7x3)). This expression involves a square root (radical), a fraction, exponents, and multiplication. To effectively simplify it, we will meticulously follow the order of operations. The initial focus will be on simplifying the expression within the radical before dealing with the square root itself. This involves applying the rules of exponents and simplifying the fraction.

Step 1: Simplifying the Fraction Inside the Radical

Following PEMDAS, we first address the expression within the radical. Our primary focus here is the fraction (63x^7)/(7x3). To simplify this fraction, we'll break it down into two parts: the numerical coefficients (63 and 7) and the variable terms (x^7 and x^3). We'll tackle each part separately and then combine the results.

Simplifying the Numerical Coefficients

The numerical part of our fraction is 63/7. This is a simple division problem. 63 divided by 7 equals 9. So, we've simplified the numerical coefficients to 9. This step demonstrates the basic arithmetic operation of division within the context of simplifying a larger expression. Accurate arithmetic is crucial for the overall correctness of the simplification process.

Simplifying the Variable Terms

Now, let's turn our attention to the variable terms: x^7/x3. Here, we employ the quotient rule of exponents. This rule states that when dividing exponents with the same base, you subtract the powers. In this case, we have x raised to the power of 7 divided by x raised to the power of 3. Applying the quotient rule, we subtract the exponents: 7 - 3 = 4. Therefore, x^7/x3 simplifies to x^4. Understanding and applying the rules of exponents is essential for simplifying algebraic expressions. The quotient rule is just one of several exponent rules that are frequently used in mathematical manipulations.

Combining the Simplified Terms

Having simplified both the numerical coefficients and the variable terms, we can now combine them. We found that 63/7 simplifies to 9, and x^7/x3 simplifies to x^4. Therefore, the entire fraction (63x^7)/(7x3) simplifies to 9x^4. This step brings together the results of our previous simplifications, demonstrating how individual components contribute to the overall simplified form of the expression. We've now successfully simplified the expression inside the radical, setting the stage for the next step: dealing with the square root.

Step 2: Evaluating the Square Root

Having simplified the fraction inside the radical to 9x^4, our expression now reads √(9x^4). The next step is to evaluate this square root. Remember, a square root asks: what value, when multiplied by itself, equals the expression under the radical? To find the square root of 9x^4, we'll consider the square root of the numerical coefficient (9) and the variable term (x^4) separately.

Finding the Square Root of the Numerical Coefficient

The square root of 9 is 3, because 3 multiplied by 3 equals 9. This is a fundamental square root that is helpful to memorize. Understanding basic square roots is essential for simplifying radical expressions. In more complex scenarios, you might need to factor the number under the radical to find perfect square factors, but in this case, 9 is a perfect square, making the process straightforward.

Finding the Square Root of the Variable Term

To find the square root of x^4, we need to recall the rules of exponents. When taking the square root of a variable raised to a power, we divide the exponent by 2. So, the square root of x^4 is x^(4/2), which simplifies to x^2. This rule is a direct consequence of the relationship between exponents and radicals. Understanding this relationship allows us to seamlessly transition between exponential and radical forms.

Combining the Results

Now we combine the square root of the numerical coefficient and the square root of the variable term. The square root of 9 is 3, and the square root of x^4 is x^2. Therefore, the square root of 9x^4 is 3x^2. This final step completes the simplification process. We've successfully evaluated the square root, arriving at the most simplified form of the expression. The result, 3x^2, is a concise and easily interpretable representation of the original expression.

Final Simplified Expression

By meticulously following the order of operations (PEMDAS) and applying the rules of exponents and radicals, we have successfully simplified the expression √((63x^7)/(7x3)). The simplified expression is:

3x^2

This result represents the most concise and simplified form of the original expression. The step-by-step approach we employed highlights the importance of order and precision in mathematical simplification. Each step built upon the previous one, ultimately leading to the final solution. The simplification process not only arrives at the correct answer but also deepens our understanding of the underlying mathematical principles.

Key Takeaways and Conclusion

Simplifying mathematical expressions requires a solid understanding of the order of operations (PEMDAS) and the rules governing exponents and radicals. This guide has demonstrated a methodical approach to simplifying the expression √((63x^7)/(7x3)), showcasing the practical application of these principles. The key takeaways from this exercise include:

• The importance of PEMDAS: Following the correct order of operations is crucial for accurate simplification.
• Simplifying within radicals first: Focus on simplifying the expression under the radical before evaluating the radical itself.
• Applying exponent rules: The quotient rule of exponents (x^m/xn = x^(m-n)) and the rule for taking the root of a power (√(x^n) = x^(n/2)) are essential tools.
• Breaking down complex expressions: Deconstructing the expression into smaller, manageable parts simplifies the process.
• Combining simplified terms: After simplifying individual components, combine them to arrive at the final simplified expression.

In conclusion, mastering the simplification of mathematical expressions is a crucial skill in mathematics and related fields. By understanding the order of operations, applying the rules of exponents and radicals, and adopting a systematic approach, you can confidently tackle even complex expressions and arrive at accurate and simplified results. The expression √((63x^7)/(7x3)), when simplified, elegantly reduces to 3x^2, demonstrating the power and precision of mathematical simplification techniques.