Simplifying Expressions Order Of Operations Example

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Simplifying expressions involving exponents can seem daunting, but by following the order of operations and understanding the rules of exponents, you can tackle even the most complex problems with confidence. This comprehensive guide will walk you through the process of simplifying the expression $\\left(7 a^6 b c6\\right){-3}$, breaking down each step and explaining the underlying principles. Mastering these concepts is crucial for success in algebra and beyond, as they form the foundation for more advanced mathematical topics. Whether you're a student looking to improve your grades or simply someone who enjoys the challenge of solving mathematical puzzles, this guide will provide you with the tools and knowledge you need to succeed. We'll start by revisiting the order of operations, often remembered by the acronym PEMDAS or BODMAS, and then delve into the specific rules of exponents that are relevant to our problem. By understanding these foundational principles, you'll be well-equipped to simplify the given expression and similar problems.

Understanding the Order of Operations (PEMDAS/BODMAS)

To effectively simplify expressions, it is critical to understand the order of operations, which dictates the sequence in which mathematical operations must be performed. This ensures that everyone arrives at the same solution for a given expression, preventing ambiguity and errors. The acronyms PEMDAS and BODMAS are commonly used to remember this order:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Following this order, we first address any expressions within parentheses or brackets. Next, we handle exponents or orders (such as square roots and cube roots). Multiplication and division are performed from left to right, followed by addition and subtraction, also from left to right. In our expression, $\\left(7 a^6 b c6\\right){-3}$, we have an expression within parentheses raised to an exponent. Therefore, we'll first focus on the exponent outside the parentheses before dealing with the terms inside. This step-by-step approach is the key to simplifying any complex expression. By consistently applying the order of operations, you can break down seemingly intimidating problems into manageable steps. Remember, accuracy in mathematics depends on adhering to these fundamental rules. So, before we dive into the specifics of our problem, ensure you're comfortable with the order of operations. It's the cornerstone of algebraic manipulation and simplification.

Applying the Power of a Product Rule

Before we proceed with simplifying the expression, let's discuss a crucial rule of exponents: 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 individually. Mathematically, this can be expressed as: $(ab)^n = a^n b^n$. This rule is essential for simplifying expressions where multiple factors are enclosed within parentheses and raised to a power, like our problem: $\\left(7 a^6 b c6\\right){-3}$. Here, we have a product of terms (7, a6a^6, b, and c6c^6) raised to the power of -3. Applying the power of a product rule, we distribute the exponent -3 to each term within the parentheses. This means we'll have 7−37^{-3}, (a6)−3(a^6)^{-3}, b−3b^{-3}, and (c6)−3(c^6)^{-3}. Understanding and applying this rule correctly is vital for simplifying expressions involving exponents. It allows us to break down a complex expression into simpler components, making it easier to manage and solve. Without this rule, simplifying such expressions would be significantly more challenging. So, make sure you grasp the power of a product rule before moving on to the next step. It's a fundamental tool in your mathematical arsenal. Remember, practice makes perfect, so try applying this rule to various expressions to solidify your understanding.

Distributing the Exponent

Now, let's apply the power of a product rule to our expression $\\left(7 a^6 b c6\\right)-3}$. As we discussed, this involves distributing the exponent -3 to each term inside the parentheses. This gives us $7^{-3 \\cdot (a6){-3} \\cdot b^{-3} \\cdot (c6){-3}$. This step is crucial because it allows us to deal with each term individually, making the simplification process more manageable. By distributing the exponent, we've transformed the original expression into a product of terms, each raised to a specific power. This is a significant step towards our final simplified form. Now, we have individual exponential terms that we can further simplify using other rules of exponents. This distribution is a key technique in simplifying complex expressions, and it's applicable in various mathematical contexts. Remember, when you encounter an expression with a product raised to a power, the first step should be to distribute the exponent to each factor within the product. This approach will almost always lead to a more straightforward simplification process. So, keep this technique in mind as you tackle other algebraic problems.

Applying the Power of a Power Rule

With the exponent distributed, we now encounter terms like (a6)−3(a^6)^{-3} and (c6)−3(c^6)^{-3}. To simplify these, we need to use another essential rule of exponents: the power of a power rule. This rule states that when a power is raised to another power, we multiply the exponents. Mathematically, this is expressed as: $(am)n = a^{m \\cdot n}$. Applying this rule to (a6)−3(a^6)^{-3}, we multiply the exponents 6 and -3, resulting in a6⋅−3=a−18a^{6 \\\cdot -3} = a^{-18}. Similarly, for (c6)−3(c^6)^{-3}, we multiply 6 and -3 to get c6⋅−3=c−18c^{6 \\\cdot -3} = c^{-18}. The power of a power rule is a fundamental concept in exponent manipulation. It allows us to simplify expressions with nested exponents, making them easier to work with. Without this rule, simplifying expressions like (a6)−3(a^6)^{-3} would be much more complex. Understanding and applying this rule correctly is vital for success in algebra and beyond. It's a tool you'll use frequently when dealing with exponents. Remember, the key is to identify when a power is raised to another power and then simply multiply the exponents. This straightforward approach will help you simplify even the most challenging expressions involving nested exponents.

Dealing with Negative Exponents

Our expression now looks like this: $7^{-3} \\cdot a^{-18} \\cdot b^{-3} \\cdot c^{-18}$. Notice that we have several terms with negative exponents. To simplify these, we need to understand how negative exponents work. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. In other words, a−n=1ana^{-n} = \\\frac{1}{a^n}. Applying this rule, we can rewrite each term with a negative exponent:

  • 7−3=1737^{-3} = \\\frac{1}{7^3}
  • a−18=1a18a^{-18} = \\\frac{1}{a^{18}}
  • b−3=1b3b^{-3} = \\\frac{1}{b^3}
  • c−18=1c18c^{-18} = \\\frac{1}{c^{18}}

Understanding negative exponents is crucial for simplifying expressions. They represent the inverse of the base raised to the positive exponent, allowing us to rewrite expressions in a more standard form. This step is often necessary to present the final answer in a simplified and conventional manner. By converting negative exponents to their reciprocal form, we eliminate the negative signs and make the expression easier to interpret. Remember, a negative exponent does not mean the value is negative; it indicates a reciprocal. This is a common misconception, so it's essential to grasp this concept thoroughly. Mastering the manipulation of negative exponents is a key skill in algebra and beyond. It allows you to express mathematical relationships in different ways and simplifies complex expressions.

Calculating the Numerical Value

We've simplified the variable terms, but we still have 7−37^{-3} to address. As we established earlier, 7−37^{-3} is equal to frac173\\\\frac{1}{7^3}. Now, we need to calculate 737^3, which means 7 multiplied by itself three times: 73=7⋅7⋅7=3437^3 = 7 \\\cdot 7 \\\cdot 7 = 343. Therefore, 7−3=13437^{-3} = \\\frac{1}{343}. Calculating the numerical value of terms like this is an essential part of simplifying expressions. It ensures that our final answer is in its most reduced form. In this case, we needed to evaluate an exponential expression, which involved repeated multiplication. Understanding how to calculate these values accurately is crucial for avoiding errors in your calculations. Remember, exponents indicate the number of times the base is multiplied by itself, and it's important to perform this calculation correctly. This step often involves simple arithmetic, but it's a critical component of the simplification process. By calculating the numerical value of 7−37^{-3}, we've taken another step closer to our final simplified expression.

Combining the Simplified Terms

Now that we've simplified each individual term, it's time to combine them to get the final simplified expression. We have:

  • 7−3=13437^{-3} = \\\frac{1}{343}
  • a−18=1a18a^{-18} = \\\frac{1}{a^{18}}
  • b−3=1b3b^{-3} = \\\frac{1}{b^3}
  • c−18=1c18c^{-18} = \\\frac{1}{c^{18}}

Multiplying these terms together, we get:

frac1343â‹…frac1a18â‹…frac1b3â‹…frac1c18=frac1343a18b3c18\\\\frac{1}{343} \\\cdot \\\\frac{1}{a^{18}} \\\cdot \\\\frac{1}{b^3} \\\cdot \\\\frac{1}{c^{18}} = \\\\frac{1}{343a^{18}b^3c^{18}}

This is our final simplified expression. Combining the simplified terms is the culmination of all the previous steps. It's where we bring together the results of each individual simplification to form the final answer. In this case, we multiplied the fractions together, resulting in a single fraction with a numerator of 1 and a denominator containing the numerical value and the variable terms with their respective exponents. This step requires careful attention to ensure that all terms are combined correctly and that the final expression is presented in its simplest form. By combining the simplified terms, we've completed the simplification process and arrived at our final answer. This step demonstrates the power of breaking down a complex problem into smaller, manageable steps and then reassembling the results to achieve the final solution.

Final Simplified Expression

Therefore, the simplified form of the expression $\\left(7 a^6 b c6\\right)-3}$ is $\\frac{1{343a{18}b3c^{18}}$. This final expression represents the original expression in its most simplified form. We've successfully applied the order of operations and the rules of exponents to achieve this result. The journey from the initial complex expression to this simplified form demonstrates the power of mathematical principles and techniques. Each step, from distributing the exponent to dealing with negative exponents and calculating numerical values, was crucial in arriving at the final answer. This simplified expression is not only more concise but also easier to understand and work with in further mathematical operations. It showcases the elegance and efficiency of mathematical simplification. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems and confidently express mathematical relationships in their simplest forms. Remember, practice is key to solidifying your understanding and improving your skills in simplifying expressions.