Solve For A Where A = (49^(2x+3) * 121^(x+5) * 343^(4x-3)) / (77^(2x+7) * 49^(7x-5)).

Exponential equations can often seem daunting, but with a systematic approach and a solid understanding of exponent rules, they can be solved effectively. In this comprehensive guide, we will dissect the equation $A = \frac{49^{2x+3} \cdot 121^{x+5} \cdot 343^{4x-3}}{77^{2x+7} \cdot 49^{7x-5}}$ step by step, revealing the underlying principles and techniques involved in finding the value of the expression. This detailed exploration is aimed at providing not just the solution, but a deep understanding of the methodology, making it easier to tackle similar problems in the future. By understanding the mechanics of solving exponential equations, you can build a strong mathematical foundation.

H2: Deconstructing the Equation: Prime Factorization

To begin this mathematical journey, the first crucial step in simplifying the equation involves expressing each term in its prime factor form. This technique allows us to break down complex numbers into their fundamental components, making it easier to manipulate and simplify the overall expression. Prime factorization is a cornerstone of number theory and is particularly useful in dealing with exponents and radicals. By rewriting each base as a power of its prime factors, we can leverage the properties of exponents to combine and simplify terms more efficiently. Let’s begin by identifying the prime factors of each base present in the equation.

• 49 can be expressed as $7^2$
• 121 can be expressed as $11^2$
• 343 can be expressed as $7^3$
• 77 can be expressed as $7 \cdot 11$

By recognizing these prime factorizations, we can rewrite the original equation in a more manageable form. This transformation is crucial for applying the rules of exponents and simplifying the expression. By expressing all bases in terms of their prime factors, we create a common ground that allows us to combine and cancel out terms, ultimately leading to a simplified solution. This step is not just about computation; it's about transforming the equation into a form where its underlying structure becomes clearer.

Now, let's substitute these prime factorizations back into the original equation. This substitution will reveal the underlying structure of the equation and set the stage for further simplification. This process of breaking down numbers into their prime factors is a fundamental technique in mathematics, and mastering it is essential for solving a wide range of problems, especially those involving exponents and radicals. The substitution process is a key step in our simplification strategy, and it will transform the equation into a form that is easier to manipulate and solve.

H2: Rewriting the Equation with Prime Factors

Now that we have identified the prime factors of each base, we can substitute these values back into the original equation. This substitution is a crucial step in simplifying the expression, as it allows us to work with common bases and apply the properties of exponents more effectively. The ability to manipulate exponential expressions is a fundamental skill in mathematics, with applications ranging from algebra to calculus and beyond. By rewriting the equation in terms of prime factors, we set the stage for further simplification and the eventual solution.

Substituting the prime factorizations, $49 = 7^2$, $121 = 11^2$, $343 = 7^3$, and $77 = 7 \cdot 11$ into the equation $A = \frac{49^{2x+3} \cdot 121^{x+5} \cdot 343^{4x-3}}{77^{2x+7} \cdot 49^{7x-5}}$, we get:

$A = \frac{(7^2)^{2x+3} \cdot (11^2)^{x+5} \cdot (7^3)^{4x-3}}{(7 \cdot 11)^{2x+7} \cdot (7^2)^{7x-5}}$

This transformation sets the stage for applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. By applying this rule, we can further simplify the exponents and consolidate the terms. This step is not just a mechanical application of a rule; it's a strategic move that simplifies the equation and brings us closer to the solution. The ability to recognize and apply the appropriate exponent rules is a hallmark of mathematical proficiency.

H2: Applying the Power of a Power Rule

Continuing our simplification process, we now apply the power of a power rule, $(a^m)^n = a^{mn}$, to the equation. This rule allows us to multiply the exponents when a power is raised to another power, which is a critical step in consolidating the terms and simplifying the expression. Understanding and applying exponent rules is fundamental to solving exponential equations, and this step demonstrates the power of these rules in action. The power of a power rule is a cornerstone of exponent manipulation, and its application here is a key step in our solution process.

Applying the rule, we get:

$A = \frac{7^{2(2x+3)} \cdot 11^{2(x+5)} \cdot 7^{3(4x-3)}}{7^{2x+7} \cdot 11^{2x+7} \cdot 7^{2(7x-5)}}$

Expanding the exponents, we have:

$A = \frac{7^{4x+6} \cdot 11^{2x+10} \cdot 7^{12x-9}}{7^{2x+7} \cdot 11^{2x+7} \cdot 7^{14x-10}}$

Now, we have expressed the equation in terms of its prime factors with simplified exponents. This form allows us to easily combine like terms, which is the next step in our simplification process. Combining like terms is a fundamental algebraic technique, and its application here will further simplify the equation, making it easier to solve. This step is not just about combining terms; it's about reorganizing the equation into a more manageable form, where the relationships between the variables and constants become clearer.

H2: Combining Like Terms with Common Bases

At this stage, our equation is expressed in terms of prime factors with simplified exponents. The next logical step is to combine like terms, specifically those with the same base. This involves applying the rule $a^m \cdot a^n = a^{m+n}$ for multiplication and $\frac{a^m}{a^n} = a^{m-n}$ for division. These rules are fundamental to manipulating exponential expressions and are crucial for simplifying the equation. Combining like terms is not just a mechanical process; it's a strategic step that consolidates the equation and brings us closer to the solution.

First, let's group the terms with the base 7 and 11:

$A = \frac{7^{4x+6} \cdot 7^{12x-9} \cdot 11^{2x+10}}{7^{2x+7} \cdot 7^{14x-10} \cdot 11^{2x+7}}$

Now, we apply the rule $a^m \cdot a^n = a^{m+n}$ to combine the exponents of 7 in the numerator:

$A = \frac{7^{(4x+6)+(12x-9)} \cdot 11^{2x+10}}{7^{2x+7} \cdot 7^{14x-10} \cdot 11^{2x+7}}$

This simplifies to:

$A = \frac{7^{16x-3} \cdot 11^{2x+10}}{7^{2x+7} \cdot 7^{14x-10} \cdot 11^{2x+7}}$

Next, we combine the exponents of 7 in the denominator:

$A = \frac{7^{16x-3} \cdot 11^{2x+10}}{7^{(2x+7)+(14x-10)} \cdot 11^{2x+7}}$

This simplifies to:

$A = \frac{7^{16x-3} \cdot 11^{2x+10}}{7^{16x-3} \cdot 11^{2x+7}}$

H2: Simplifying the Expression by Cancelling Terms

Having combined the like terms, we now have a simplified equation where we can clearly see common factors in the numerator and the denominator. The next step is to cancel out these common factors, which will further reduce the complexity of the expression. Cancelling common factors is a fundamental algebraic technique that simplifies equations and reveals their underlying structure. This step is not just about removing terms; it's about making the equation more transparent and easier to analyze.

We observe that $7^{16x-3}$ appears in both the numerator and the denominator, so we can cancel it out:

$A = \frac{7^{16x-3} \cdot 11^{2x+10}}{7^{16x-3} \cdot 11^{2x+7}}$

Cancelling the common factor, we get:

$A = \frac{11^{2x+10}}{11^{2x+7}}$

Now, we can apply the rule $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression further:

$A = 11^{(2x+10)-(2x+7)}$

Simplifying the exponent, we have:

$A = 11^{2x+10-2x-7}$

$A = 11^3$

H2: Final Calculation and Solution

After simplifying the expression, we have arrived at a point where we can easily calculate the final value of A. The expression has been reduced to a simple power of 11, which can be directly computed. This final calculation is the culmination of our simplification efforts and provides the solution to the original equation. The ability to arrive at a clear and concise solution is a testament to the power of systematic problem-solving techniques.

Now, we just need to calculate $11^3$:

$A = 11^3 = 11 \cdot 11 \cdot 11 = 121 \cdot 11 = 1331$

Therefore, the value of A is 1331.

H2: Conclusion: Mastering Exponential Equations

In this comprehensive guide, we have successfully navigated the complexities of the exponential equation $A = \frac{49^{2x+3} \cdot 121^{x+5} \cdot 343^{4x-3}}{77^{2x+7} \cdot 49^{7x-5}}$. By breaking down the problem into manageable steps, we have demonstrated the power of prime factorization, exponent rules, and simplification techniques. This detailed exploration not only provides the solution but also equips you with the skills and understanding necessary to tackle similar problems with confidence. The journey through this equation highlights the importance of a systematic approach to mathematical problem-solving.

The key takeaways from this exploration include:

• Prime Factorization: Expressing numbers in terms of their prime factors is a powerful technique for simplifying exponential expressions.
• Exponent Rules: Understanding and applying the rules of exponents, such as the power of a power rule and the quotient rule, is crucial for simplification.
• Combining Like Terms: Grouping and combining like terms allows for further simplification and reduction of the expression.
• Cancelling Common Factors: Identifying and cancelling common factors in the numerator and denominator simplifies the equation.

By mastering these techniques, you can confidently approach exponential equations and other mathematical challenges. The ability to solve complex equations is not just about finding the answer; it's about developing a deep understanding of the underlying principles and building a strong foundation for further mathematical exploration.