Simplifying Expressions 36x^14 Divided By 3x^5 Using Exponent Rules

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In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent complex equations in a more concise and manageable form. When dealing with expressions involving division and exponents, a clear understanding of the properties of these operations is essential. This article delves into the process of simplifying expressions of the form (36x^{14}) ÷ (3x^{5}), employing the properties of division and exponents to arrive at the simplest form.

The expression (36x^14}) ÷ (3x^{5}) represents the division of two terms 36x^{14 and 3x^{5}. The goal is to simplify this expression by applying the rules of division and exponents. The key properties that come into play here are the quotient of powers property and the division of coefficients.

Understanding the Properties

Before we dive into the simplification process, let's first understand the properties we'll be using:

  1. Quotient of Powers Property: This property states that when dividing exponents with the same base, you subtract the exponents. Mathematically, it can be expressed as: x^{m} / x^{n} = x^{m-n}, where x is the base and m and n are the exponents.

  2. Division of Coefficients: When dividing terms with coefficients, you simply divide the coefficients as you would with any numerical division. For example, (36a) / (3b) would simplify to (36/3) * (a/b) = 12 * (a/b).

Applying the Properties to Simplify the Expression

Now that we have a grasp of the properties, let's apply them to our expression (36x^{14}) ÷ (3x^{5}).

Step 1: Separate the Coefficients and Variables

First, we can separate the coefficients (the numerical parts) and the variables (the parts with exponents) of the expression:

(36x^{14}) ÷ (3x^{5}) = (36 ÷ 3) * (x^{14} ÷ x^{5})

This separation allows us to deal with the numerical division and the variable division separately, making the simplification process clearer.

Step 2: Divide the Coefficients

Next, we perform the division of the coefficients:

36 ÷ 3 = 12

This simplifies the numerical part of our expression to 12.

Step 3: Apply the Quotient of Powers Property

Now, we turn our attention to the variables and their exponents. We have x^{14} ÷ x^{5}. According to the quotient of powers property, we subtract the exponents:

x^{14} ÷ x^{5} = x^{14-5} = x^{9}

This simplifies the variable part of our expression to x^{9}.

Step 4: Combine the Simplified Parts

Finally, we combine the simplified coefficient and variable parts:

(36 ÷ 3) * (x^{14} ÷ x^{5}) = 12 * x^{9} = 12x^{9}

Therefore, the simplified form of the expression (36x^{14}) ÷ (3x^{5}) is 12x^{9}.

Example Walkthrough: Simplifying (36x^{14}) ÷ (3x^{5})

Let’s break down the simplification of the expression (36x^{14}) ÷ (3x^{5}) step by step. This walkthrough will reinforce your understanding of how to apply the properties of division and exponents effectively. Understanding the properties of exponents is crucial for simplifying complex mathematical expressions.

Step 1: Separate the Coefficients and Variables

When faced with an expression like (36x^{14}) ÷ (3x^{5}), the initial step involves separating the coefficients (the numerical parts) from the variables (the parts with exponents). This separation is based on the fundamental principle that multiplication and division can be rearranged due to their associative and commutative properties. By isolating the coefficients and variables, we can apply the appropriate rules more clearly and efficiently. This preparatory step sets the stage for a more organized simplification process. Separating coefficients and variables allows for focused application of mathematical rules.

(36x^{14}) ÷ (3x^{5}) = (36 ÷ 3) × (x^{14} ÷ x^{5})

Step 2: Divide the Coefficients

After separating the coefficients, the next logical step is to perform the division of these numerical values. In this case, we have 36 ÷ 3, which is a straightforward arithmetic operation. Dividing 36 by 3 gives us 12. This result becomes the numerical coefficient of the simplified expression. The simplicity of this step highlights the importance of separating coefficients, as it allows us to focus on basic arithmetic without the distraction of variables and exponents. Performing division of coefficients is a fundamental arithmetic operation.

36 ÷ 3 = 12

Step 3: Apply the Quotient of Powers Property

With the coefficients simplified, we now turn our attention to the variables and their exponents. The expression x^{14} ÷ x^{5} involves dividing two powers with the same base (x). Here, the quotient of powers property comes into play. This property states that when dividing exponents with the same base, you subtract the exponents. Applying this rule, we subtract the exponent in the denominator (5) from the exponent in the numerator (14). This step is crucial for simplifying expressions involving exponents and demonstrates a key principle in algebraic manipulation. The quotient of powers property simplifies division of exponents with the same base.

x^{14} ÷ x^{5} = x^{14-5} = x^{9}

Step 4: Combine the Simplified Parts

Having simplified both the coefficients and the variables with exponents, the final step is to combine these simplified parts back into a single expression. We have found that the coefficients simplify to 12 and the variables simplify to x^{9}. Multiplying these two results together gives us the fully simplified form of the original expression. This step demonstrates the culmination of the simplification process, where individual components are brought together to form the final, concise expression. Combining simplified parts yields the final simplified expression.

(36 ÷ 3) × (x^{14} ÷ x^{5}) = 12 × x^{9} = 12x^{9}

Thus, the simplified form of the expression (36x^{14}) ÷ (3x^{5}) is 12x^{9}. This step-by-step walkthrough illustrates the systematic application of mathematical properties to simplify complex expressions. By breaking down the process into smaller, manageable steps, we can clearly see how each property contributes to the final result. Step-by-step simplification enhances understanding of mathematical processes.

Additional Examples

To solidify your understanding, let's look at a couple more examples:

Example 1: Simplify (25y^{8}) ÷ (5y^{2})

  1. Separate coefficients and variables: (25 ÷ 5) * (y^{8} ÷ y^{2})
  2. Divide coefficients: 25 ÷ 5 = 5
  3. Apply quotient of powers property: y^{8} ÷ y^{2} = y^{8-2} = y^{6}
  4. Combine simplified parts: 5 * y^{6} = 5y^{6}

Therefore, the simplified form of (25y^{8}) ÷ (5y^{2}) is 5y^{6}.

Example 2: Simplify (48z^{11}) ÷ (12z^{4})

  1. Separate coefficients and variables: (48 ÷ 12) * (z^{11} ÷ z^{4})
  2. Divide coefficients: 48 ÷ 12 = 4
  3. Apply quotient of powers property: z^{11} ÷ z^{4} = z^{11-4} = z^{7}
  4. Combine simplified parts: 4 * z^{7} = 4z^{7}

Therefore, the simplified form of (48z^{11}) ÷ (12z^{4}) is 4z^{7}.

Common Mistakes to Avoid

While simplifying expressions with division and exponents, it's crucial to be aware of common mistakes that students often make. Avoiding common mistakes ensures accuracy in mathematical simplification.

Mistake 1: Incorrectly Applying the Quotient of Powers Property

One frequent error is misapplying the quotient of powers property. Students may mistakenly add the exponents instead of subtracting them when dividing terms with the same base. Remember, the property states that x^{m} / x^{n} = x^{m-n}, so you should always subtract the exponents. Misapplication of the quotient of powers property leads to incorrect simplification.

Correct: x^{5} / x^{2} = x^{5-2} = x^{3}

Incorrect: x^{5} / x^{2} = x^{5+2} = x^{7}

Mistake 2: Forgetting to Simplify Coefficients

Another common mistake is overlooking the simplification of coefficients. It's essential to divide the coefficients separately before dealing with the variables and exponents. Neglecting this step can lead to an incomplete simplification. Forgetting to simplify coefficients results in an incomplete answer.

Correct: (18x^{7}) / (6x^{3}) = (18 / 6) * (x^{7} / x^{3}) = 3x^{4}

Incorrect: (18x^{7}) / (6x^{3}) = x^{4} (missing the division of 18 by 6)

Mistake 3: Mixing Up Quotient and Product Rules

Students sometimes confuse the quotient of powers property with the product of powers property. The product of powers property states that when multiplying exponents with the same base, you add the exponents (x^{m} * x^{n} = x^{m+n}). It’s crucial to differentiate between these two properties and apply the correct one based on the operation (division or multiplication). Confusing quotient and product rules leads to errors in simplification.

Correct (Quotient): x^{5} / x^{2} = x^{5-2} = x^{3}

Correct (Product): x^{5} * x^{2} = x^{5+2} = x^{7}

Mistake 4: Neglecting Negative Exponents

When applying the quotient of powers property, you might encounter negative exponents. For example, if you have x^{2} / x^{5}, subtracting the exponents gives you x^{2-5} = x^{-3}. Remember that a negative exponent means you should take the reciprocal of the base raised to the positive exponent (x^{-n} = 1 / x^{n}). Neglecting negative exponents results in an incomplete solution.

Correct: x^{2} / x^{5} = x^{2-5} = x^{-3} = 1 / x^{3}

Incorrect: x^{2} / x^{5} = x^{-3} (leaving the answer with a negative exponent)

Mistake 5: Ignoring the Order of Operations

Failing to follow the correct order of operations (PEMDAS/BODMAS) can also lead to mistakes. Make sure to perform the division of coefficients and the subtraction of exponents in the correct order. Ignoring the order of operations can lead to calculation errors.

By being mindful of these common mistakes and practicing regularly, you can improve your accuracy in simplifying expressions with division and exponents. Regular practice is key to mastering simplification techniques.

Conclusion

Simplifying expressions involving division and exponents is a vital skill in mathematics. By understanding and applying the quotient of powers property and the rules for dividing coefficients, we can effectively reduce complex expressions to their simplest forms. Remember to separate the coefficients and variables, apply the quotient of powers property correctly, and combine the simplified parts. With practice, you'll become proficient at simplifying these types of expressions. Mastering simplification techniques enhances mathematical proficiency.

By avoiding common mistakes and consistently applying these steps, you can confidently simplify a wide range of expressions involving division and exponents. Consistent application of steps leads to accurate simplification. This skill is not only essential for academic success but also for various real-world applications where mathematical expressions need to be simplified for practical use. Real-world applications benefit from simplified mathematical expressions.