Solving (15)^{10} X (10)^{10} = (...)^2 A Step-by-Step Guide

Introduction: Decoding the Mathematical Puzzle

In this article, we delve into the mathematical equation (15)^{10} \times (10)^{10} = (...)^2, aiming to decipher the unknown value that satisfies the equation. This problem involves understanding the properties of exponents and how they interact with multiplication and squaring. To solve this, we will break down the equation step by step, utilizing fundamental mathematical principles to arrive at the solution. The beauty of mathematics lies in its ability to simplify complex problems into manageable steps, and this equation is a perfect example of that. Through careful manipulation and application of exponent rules, we will transform the left-hand side of the equation into a form that clearly reveals the value that, when squared, equals the initial expression. This journey will not only provide the answer but also enhance our understanding of mathematical operations and problem-solving strategies. Let's embark on this mathematical adventure and unlock the mystery behind this equation.

Breaking Down the Equation: Exponents and Multiplication

To solve the equation (15)^{10} \times (10)^{10} = (...)^2, our first step is to understand how exponents behave when numbers are multiplied. A key property of exponents states that when multiplying numbers with the same exponent, we can multiply the bases while keeping the exponent the same. Mathematically, this is expressed as a^n \times b^n = (a \times b)^n. Applying this rule to our equation, we can combine the terms on the left-hand side. This involves recognizing that both (15)^{10} and (10)^{10} have the same exponent, which is 10. Therefore, we can multiply the bases, 15 and 10, together while maintaining the exponent of 10. This transformation simplifies the equation and brings us closer to finding the solution. It's a crucial step because it consolidates the two terms into a single term, making it easier to manipulate and compare with the right-hand side of the equation. This process highlights the power of mathematical rules in simplifying complex expressions and paving the way for a clearer path to the solution. Understanding and applying these rules is fundamental to solving not just this equation, but a wide range of mathematical problems involving exponents.

Simplifying the Expression: Combining the Bases

Following the exponent rule, let's apply it to our equation (15)^{10} \times (10)^{10} = (...)^2. Multiplying the bases 15 and 10, we get 150. So, the left-hand side of the equation becomes (15 \times 10)^{10} = (150)^{10}. This simplification is a significant step forward. We've successfully combined the two original terms into a single term raised to the power of 10. This makes the equation much cleaner and easier to work with. Now, our equation looks like (150)^{10} = (...)^2. The next challenge is to figure out what number, when squared, will give us (150)^{10}. This requires us to think about the relationship between exponents and squares. We need to find a way to express (150)^{10} as a perfect square, which means it should be in the form of something raised to the power of 2. This involves understanding how exponents interact with each other, particularly when dealing with powers of powers. The ability to simplify expressions like this is a crucial skill in algebra and is essential for solving more complex mathematical problems.

Expressing as a Square: Understanding Powers of Powers

Now that we have the equation (150)^10} = (...)^2**, we need to express the left-hand side as a square. To do this, we can use another important property of exponents **(a^m)n = a^{m \times n. This rule tells us that raising a power to another power is the same as multiplying the exponents. In our case, we want to express (150)^{10} as something squared, so we need to find a way to write the exponent 10 as a product where one of the factors is 2. We can easily do this by recognizing that 10 is equal to 5 multiplied by 2 (10 = 5 \times 2). Therefore, we can rewrite (150)^{10} as (150)^{5 \times 2}. Now, applying the power of a power rule, we can rewrite this as ((150)⁵⁾2. This transformation is the key to solving the equation. We have successfully expressed the left-hand side as a perfect square. This step demonstrates the elegance and power of exponent rules in manipulating mathematical expressions. By understanding and applying these rules, we can transform complex equations into simpler, more manageable forms.

Finding the Solution: Unveiling the Unknown

With the equation now in the form ((150)⁵⁾2 = (...)^2, the solution becomes clear. We have successfully expressed the left-hand side as a square, specifically as the square of (150)^5. Therefore, the value that fits into the parentheses on the right-hand side is simply (150)^5. This means that (150)^5 squared is equal to (15)^{10} \times (10)^{10}. To calculate the actual value of (150)^5, we would multiply 150 by itself five times. This would result in a very large number, but the important thing is that we have found the expression that, when squared, satisfies the original equation. This final step highlights the importance of understanding the properties of exponents and how they can be used to solve mathematical problems. By breaking down the equation into smaller, more manageable steps, and by applying the appropriate exponent rules, we were able to unravel the puzzle and arrive at the solution. This process demonstrates the power of mathematical reasoning and problem-solving skills.

Conclusion: Mastering Exponents and Equations

In conclusion, we have successfully solved the equation (15)^{10} \times (10)^{10} = (...)^2 by systematically applying the rules of exponents. We first combined the terms on the left-hand side by multiplying the bases while keeping the exponent the same, resulting in (150)^{10}. Then, we expressed this as a square by recognizing that 10 can be written as 5 times 2, allowing us to rewrite the expression as ((150)⁵⁾2. This revealed that the solution to the equation is (150)^5. This exercise has not only provided us with the answer but has also reinforced our understanding of how exponents work and how they can be manipulated to solve equations. Mastering these concepts is crucial for success in mathematics and related fields. The ability to break down complex problems into smaller, more manageable steps, and to apply the appropriate rules and formulas, is a valuable skill that can be applied to a wide range of challenges. As we continue our mathematical journey, the principles and techniques learned in this exercise will serve as a solid foundation for tackling more advanced problems and concepts.