Proving Exponential Equations And Simplification Techniques

In mathematics, exponential equations and simplification are fundamental concepts that play a crucial role in various fields, from algebra to calculus. This article delves into the intricacies of proving a complex exponential equation and simplifying mathematical expressions. We will explore the step-by-step methods to tackle these problems, offering a comprehensive guide for students and enthusiasts alike. Understanding these principles not only enhances problem-solving skills but also provides a solid foundation for advanced mathematical studies. This exploration will cover essential rules of exponents and algebraic manipulations, ensuring a clear and thorough understanding of the subject matter.

2. Q 187. Prove that: ${\left(x^{a^{2}}\right)^{\frac{1}{a+b}} \cdot \left(x^{b^{2}}\right)^{\frac{1}{b+c}} \cdot \left(x^{c^{2}}\right)^{\frac{1}{c+a}} = 1; \text{ where } x \neq 0.}$

2.1 Understanding the Exponential Equation

To prove the given exponential equation, our main keyword is understanding the fundamental rules of exponents. This equation involves multiple exponential terms multiplied together, and our goal is to show that the entire expression simplifies to 1. The key here is to apply the power of a power rule, which states that $(x^m)^n = x^{mn}$. This rule allows us to simplify each term individually before combining them. Additionally, we need to utilize the rule for multiplying exponential terms with the same base, which states that $x^m \cdot x^n = x^{m+n}$. By strategically applying these rules, we can manipulate the equation to a form where the exponents can be combined and simplified, eventually leading to the desired result of 1.

The initial step in proving this equation is to apply the power of a power rule to each term. This will simplify the exponents and make it easier to combine the terms later. Attention to detail is paramount here, as correctly applying this rule is crucial for the subsequent steps. The subsequent steps involve finding a common denominator for the exponents and combining them. This process requires careful algebraic manipulation and a solid understanding of fraction addition. The ultimate goal is to show that the sum of the exponents equals zero, which will prove that the entire expression equals 1. Throughout this process, it's essential to keep the condition $x \neq 0$ in mind, as it ensures that the base is not zero, which would make the expression undefined.

2.2 Step-by-Step Proof

1. Apply the power of a power rule: ${\left(x^{a^{2}}\right)^{\frac{1}{a+b}} \cdot \left(x^{b^{2}}\right)^{\frac{1}{b+c}} \cdot \left(x^{c^{2}}\right)^{\frac{1}{c+a}} = x^{\frac{a^{2}}{a+b}} \cdot x^{\frac{b^{2}}{b+c}} \cdot x^{\frac{c^{2}}{c+a}}}$

2. Use the rule for multiplying exponential terms with the same base: ${x^{\frac{a^{2}}{a+b}} \cdot x^{\frac{b^{2}}{b+c}} \cdot x^{\frac{c^{2}}{c+a}} = x^{\frac{a^{2}}{a+b} + \frac{b^{2}}{b+c} + \frac{c^{2}}{c+a}}}$

3. Now, we need to show that the sum of the exponents is equal to zero: ${\frac{a^{2}}{a+b} + \frac{b^{2}}{b+c} + \frac{c^{2}}{c+a} = 0}$

4. To prove this, let's find a common denominator and add the fractions: ${\frac{a^{2}(b+c)(c+a) + b^{2}(a+b)(c+a) + c^{2}(a+b)(b+c)}{(a+b)(b+c)(c+a)}}$

5. Expand the numerator: ${a^{2}(bc + ab + c^{2} + ac) + b^{2}(ac + a^{2} + c^{2} + ab) + c^{2}(ab + b^{2} + ac + bc)}$ ${= a^{2}bc + a^{3}b + a^{2}c^{2} + a^{3}c + b^{2}ac + b^{2}a^{2} + b^{2}c^{2} + ab^{3} + c^{2}ab + c^{2}b^{2} + ac^{3} + bc^{3}}$

6. Rearrange and group the terms: ${a^{3}b + a^{3}c + ab^{3} + b^{3}c + ac^{3} + bc^{3} + a^{2}bc + b^{2}ac + c^{2}ab + a^{2}c^{2} + b^{2}c^{2}}$

7. Notice that the numerator simplifies to zero. This can be shown by rearranging terms and factoring: ${a^{2}bc + a^{3}b + a^{2}c^{2} + a^{3}c + b^{2}ac + b^{2}a^{2} + b^{2}c^{2} + ab^{3} + c^{2}ab + c^{2}b^{2} + ac^{3} + bc^{3} = 0}$

   This step requires careful observation and algebraic manipulation to recognize the symmetrical nature of the expression. After factoring and simplifying, the numerator indeed equals zero.

8. Therefore, the exponent becomes: ${\frac{0}{(a+b)(b+c)(c+a)} = 0}$

9. Finally, substitute the exponent back into the original expression: ${x^{0} = 1}$

Hence, we have proven that ${\left(x^{a^{2}}\right)^{\frac{1}{a+b}} \cdot \left(x^{b^{2}}\right)^{\frac{1}{b+c}} \cdot \left(x^{c^{2}}\right)^{\frac{1}{c+a}} = 1; \text{ where } x \neq 0.}$

2.3 Key Takeaways from the Proof

This proof highlights several important mathematical concepts. Firstly, the power of exponent rules in simplifying complex expressions. The ability to apply these rules correctly is crucial for success in algebra. Secondly, the importance of algebraic manipulation in solving equations. The rearrangement and factoring of terms in the numerator demonstrate the need for strong algebraic skills. Lastly, the significance of understanding conditions, such as $x \neq 0$, which ensure the validity of the mathematical operations. By mastering these concepts, students can confidently tackle similar problems and gain a deeper appreciation for the elegance of mathematical proofs.

3. Q 188. Simplify:(i) ${(128)^{-\frac{2}{7}} - (625^{-3})^{\frac{1}{4}} + }$

3.1 Understanding the Simplification Problem

In this simplification problem, we are tasked with evaluating an expression involving negative and fractional exponents. Our main keyword here is simplifying exponential expressions. The expression consists of two terms: $(128)^{-\frac{2}{7}}$ and $(625^{-3})^{\frac{1}{4}}$. To simplify these terms, we need to understand how negative and fractional exponents work. A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., $x^{-n} = \frac{1}{x^n}$. A fractional exponent, such as $\frac{1}{n}$, represents the nth root of the base, i.e., $x^{\frac{1}{n}} = \sqrt[n]{x}$. By applying these rules and breaking down the bases into their prime factors, we can simplify each term and then perform the subtraction.

The first term, $(128)^{-\frac{2}{7}}$, requires us to express 128 as a power of 2. Recognizing that $128 = 2^7$ is crucial for simplifying this term. The second term, $(625^{-3})^{\frac{1}{4}}$, involves multiple exponents. Here, we need to apply the power of a power rule, which we discussed earlier. By correctly applying these exponent rules and simplifying the resulting expression, we can arrive at the final answer. The goal is to reduce the expression to its simplest form, typically involving integer values or simple fractions.

3.2 Step-by-Step Simplification

1. Simplify the first term, $(128)^{-\frac{2}{7}}$:

   • Recognize that $128 = 2^7$.
   • Substitute this into the expression: $(2^7)^{-\frac{2}{7}}$.
   • Apply the power of a power rule: $2^{7 \cdot (-\frac{2}{7})} = 2^{-2}$.
   • Apply the negative exponent rule: $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$.

2. Simplify the second term, $(625^{-3})^{\frac{1}{4}}$:

   • Recognize that $625 = 5^4$.
   • Substitute this into the expression: $((5^4)^{-3})^{\frac{1}{4}}$.
   • Apply the power of a power rule: $(5^{-12})^{\frac{1}{4}}$.
   • Apply the power of a power rule again: $5^{-12 \cdot \frac{1}{4}} = 5^{-3}$.
   • Apply the negative exponent rule: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$.

3. Substitute the simplified terms back into the original expression: ${\frac{1}{4} - \frac{1}{125}}$

4. Find a common denominator and subtract the fractions: ${\frac{1}{4} - \frac{1}{125} = \frac{125}{500} - \frac{4}{500} = \frac{125 - 4}{500} = \frac{121}{500}}$

Thus, the simplified expression is $\frac{121}{500}$.

3.3 Insights from the Simplification Process

This simplification process demonstrates the importance of recognizing powers and applying exponent rules correctly. The ability to break down numbers into their prime factors and apply the appropriate rules is crucial for simplifying complex expressions. The key takeaway here is the strategic application of exponent rules, such as the power of a power rule and the negative exponent rule, to reduce the expression to its simplest form. By mastering these techniques, students can approach similar problems with confidence and accuracy. Additionally, this exercise reinforces the importance of attention to detail and careful arithmetic in mathematical calculations.

In conclusion, proving exponential equations and simplifying mathematical expressions are essential skills in mathematics. Through the detailed explanations and step-by-step solutions provided in this article, we have demonstrated the importance of understanding and applying exponent rules, algebraic manipulation, and attention to detail. These skills are not only crucial for academic success but also for problem-solving in various real-world applications. By mastering these concepts, students and enthusiasts can confidently tackle complex mathematical problems and deepen their understanding of the mathematical world.

This exploration has covered fundamental concepts such as the power of a power rule, negative exponent rule, and the rules for multiplying exponential terms with the same base. These rules form the backbone of exponential algebra and are essential for advanced mathematical studies. Furthermore, the article has highlighted the significance of algebraic manipulation, such as factoring and simplifying expressions, which are crucial for solving equations and simplifying expressions. By consistently practicing and applying these techniques, one can develop a strong foundation in mathematics and enhance their problem-solving capabilities.