Simplifying Exponential Expressions A = (7^20 × 7^8) / (7^10 × 7^2)

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In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying an exponential expression, specifically focusing on the expression A = (7^20 × 7^8) / (7^10 × 7^2). We will break down the steps involved, explain the underlying mathematical principles, and provide a clear, comprehensive guide to help you master the art of simplifying exponential expressions.

Understanding Exponential Expressions

Before we dive into the simplification process, it's crucial to grasp the concept of exponential expressions. An exponential expression consists of a base and an exponent. The base is the number being multiplied, and the exponent indicates how many times the base is multiplied by itself. For instance, in the expression 7^20, 7 is the base, and 20 is the exponent, meaning 7 is multiplied by itself 20 times.

Exponential expressions are governed by a set of rules known as the laws of exponents. These laws provide the foundation for simplifying and manipulating exponential expressions. The key laws that are relevant to our problem are:

  1. Product of Powers: When multiplying exponents with the same base, add the powers. Mathematically, this is expressed as: a^m × a^n = a^(m+n)
  2. Quotient of Powers: When dividing exponents with the same base, subtract the powers. This is represented as: a^m / a^n = a^(m-n)

With these laws in mind, we can now embark on the journey of simplifying the expression A = (7^20 × 7^8) / (7^10 × 7^2).

Step-by-Step Simplification

Let's dissect the expression step by step, applying the laws of exponents to arrive at the simplified form.

Step 1: Simplify the Numerator

The numerator of our expression is 7^20 × 7^8. According to the product of powers rule, when multiplying exponents with the same base, we add the powers. Therefore:

7^20 × 7^8 = 7^(20+8) = 7^28

Step 2: Simplify the Denominator

Now, let's focus on the denominator, which is 7^10 × 7^2. Again, we apply the product of powers rule:

7^10 × 7^2 = 7^(10+2) = 7^12

Step 3: Simplify the Entire Expression

Having simplified both the numerator and the denominator, our expression now looks like this:

A = 7^28 / 7^12

To simplify further, we invoke the quotient of powers rule, which states that when dividing exponents with the same base, we subtract the powers. Hence:

A = 7^(28-12) = 7^16

Therefore, the simplified form of the expression A = (7^20 × 7^8) / (7^10 × 7^2) is 7^16.

Detailed Explanation and Practical Examples

Let's dive deeper into each step of the simplification process, providing more detailed explanations and practical examples to solidify your understanding.

In-Depth Look at the Product of Powers Rule

The product of powers rule (a^m × a^n = a^(m+n)) is a cornerstone of exponential expression simplification. It essentially states that when you multiply two exponential terms with the same base, you can simplify the expression by adding their exponents. This rule stems from the fundamental definition of exponents as repeated multiplication.

Consider the example of 2^3 × 2^2. Expanding this, we get (2 × 2 × 2) × (2 × 2). We are multiplying 2 by itself a total of 5 times, which can be written as 2^5. Notice that 5 is the sum of the original exponents, 3 and 2. This illustrates the logic behind the product of powers rule.

To further illustrate this, consider the original expression: 7^20 × 7^8.

  • 7^20 means 7 multiplied by itself 20 times.
  • 7^8 means 7 multiplied by itself 8 times.

When you multiply these two terms together, you are essentially multiplying 7 by itself a total of 20 + 8 = 28 times. Hence, 7^20 × 7^8 = 7^28.

Understanding the Quotient of Powers Rule

The quotient of powers rule (a^m / a^n = a^(m-n)) complements the product of powers rule. It states that when you divide two exponential terms with the same base, you can simplify the expression by subtracting their exponents. This rule is also rooted in the definition of exponents and the properties of division.

Take, for instance, the expression 3^5 / 3^2. Expanding this, we have (3 × 3 × 3 × 3 × 3) / (3 × 3). We can cancel out two 3s from both the numerator and the denominator, leaving us with 3 × 3 × 3, which is 3^3. Notice that 3 is the difference between the original exponents, 5 and 2. This exemplifies the rationale behind the quotient of powers rule.

In our problem, we had 7^28 / 7^12. This means we are dividing 7 multiplied by itself 28 times by 7 multiplied by itself 12 times. We can cancel out 12 factors of 7 from both the numerator and the denominator, leaving us with 7 multiplied by itself 28 - 12 = 16 times. Therefore, 7^28 / 7^12 = 7^16.

Putting it All Together: A Practical Example

To solidify your grasp of these concepts, let's consider another example:

Simplify the expression: B = (5^12 × 5^5) / (5^3 × 5^2)

  1. Simplify the Numerator:

    5^12 × 5^5 = 5^(12+5) = 5^17 (Applying the product of powers rule)

  2. Simplify the Denominator:

    5^3 × 5^2 = 5^(3+2) = 5^5 (Applying the product of powers rule)

  3. Simplify the Entire Expression:

    B = 5^17 / 5^5 = 5^(17-5) = 5^12 (Applying the quotient of powers rule)

Therefore, the simplified form of the expression B is 5^12.

Common Mistakes to Avoid

While simplifying exponential expressions, it's easy to fall prey to common mistakes. Being aware of these pitfalls can help you avoid them and ensure accuracy in your calculations.

  1. Incorrectly Applying the Product of Powers Rule: A common mistake is to multiply the bases when applying the product of powers rule. Remember, this rule applies only when the bases are the same. For example, 2^3 × 2^2 is not equal to 4^5. The correct application is 2^3 × 2^2 = 2^(3+2) = 2^5.

  2. Incorrectly Applying the Quotient of Powers Rule: Similar to the product of powers rule, a frequent error is to divide the bases when applying the quotient of powers rule. Again, this rule is valid only when the bases are the same. For instance, 3^5 / 3^2 is not equal to 1^3. The correct application is 3^5 / 3^2 = 3^(5-2) = 3^3.

  3. Forgetting to Simplify Completely: After applying the laws of exponents, ensure that you have simplified the expression as much as possible. This may involve further applications of the rules or evaluating the final exponential term.

  4. Ignoring Order of Operations: When simplifying complex expressions, remember to adhere to the order of operations (PEMDAS/BODMAS). Exponents should be evaluated before multiplication, division, addition, or subtraction.

Conclusion: Mastering Exponential Expression Simplification

Simplifying exponential expressions is a crucial skill in mathematics. By understanding the laws of exponents, practicing step-by-step simplification, and being mindful of common mistakes, you can master this skill and confidently tackle complex mathematical problems. The expression A = (7^20 × 7^8) / (7^10 × 7^2), which simplifies to 7^16, serves as a prime example of how the product and quotient of powers rules can be effectively applied to simplify exponential expressions. Keep practicing, and you'll become proficient in the art of simplifying exponential expressions.