Simplify 30√(45/25) A Step By Step Guide
Understanding the Problem
In this mathematical problem, we are tasked with simplifying the expression 30√(45/25). This involves several key steps, including understanding square roots, simplifying fractions, and performing arithmetic operations. Our goal is to break down the expression into its simplest form, making it easier to understand and work with. This type of problem often appears in algebra and pre-calculus courses, where students are learning to manipulate and simplify algebraic expressions. Simplifying expressions is a fundamental skill in mathematics, as it allows us to solve more complex problems and understand underlying mathematical concepts more clearly.
To approach this problem effectively, we need to remember some core mathematical principles. First, understanding square roots is crucial. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Second, we need to know how to simplify fractions. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. For example, the fraction 4/6 can be simplified to 2/3 by dividing both the numerator and the denominator by 2. Third, we must be comfortable with arithmetic operations such as multiplication and division. Finally, we should remember the order of operations (PEMDAS/BODMAS), which tells us to perform operations in the following order: parentheses/brackets, exponents/orders, multiplication and division (from left to right), and addition and subtraction (from left to right).
Let's start by outlining the steps we will take to solve this problem. First, we will simplify the fraction inside the square root. This involves finding the greatest common divisor of the numerator and the denominator and dividing both by it. Second, we will simplify the square root itself, if possible. This might involve breaking down the number inside the square root into its prime factors and looking for pairs of factors that can be taken out of the square root. Third, we will perform any remaining arithmetic operations, such as multiplication, to arrive at our final simplified answer. By following these steps, we can systematically simplify the expression and arrive at the correct solution. Each step builds upon the previous one, so it is important to approach the problem in a logical and organized manner. This not only helps us find the correct answer but also improves our understanding of the underlying mathematical principles.
Step-by-Step Solution
To simplify the expression 30√(45/25), we will proceed step by step, ensuring each operation is performed correctly.
Step 1: Simplify the Fraction Inside the Square Root
The first step is to simplify the fraction inside the square root, which is 45/25. To simplify this fraction, we need to find the greatest common divisor (GCD) of 45 and 25. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 25 are 1, 5, and 25. The greatest common divisor of 45 and 25 is 5. Now, we divide both the numerator and the denominator by 5:
45 ÷ 5 = 9
25 ÷ 5 = 5
So, the simplified fraction is 9/5. This means our expression now looks like this:
30√(9/5)
Simplifying the fraction inside the square root is a crucial step because it makes the subsequent steps easier to manage. By reducing the numbers inside the square root, we are working with smaller values, which simplifies calculations and reduces the risk of errors. This initial simplification also helps in recognizing perfect squares, which can be easily taken out of the square root, further simplifying the expression. Understanding how to simplify fractions is a fundamental skill in mathematics, and it is applied in various contexts, including algebra, calculus, and even real-world applications such as cooking and finance. Mastering this skill ensures a solid foundation for more advanced mathematical concepts.
Step 2: Simplify the Square Root
Next, we need to simplify the square root √(9/5). We can rewrite this as the square root of the numerator divided by the square root of the denominator:
√(9/5) = √9 / √5
The square root of 9 is 3, so we have:
√9 = 3
Thus, our expression becomes:
3 / √5
Now, we have 3 / √5. To rationalize the denominator, we multiply both the numerator and the denominator by √5:
(3 / √5) * (√5 / √5) = (3√5) / 5
So, √(9/5) simplifies to (3√5) / 5. The process of rationalizing the denominator is important because it removes the square root from the denominator, which is a common convention in mathematics. It also makes it easier to compare and combine terms involving radicals. Rationalizing the denominator involves multiplying both the numerator and the denominator by the square root in the denominator, which effectively eliminates the square root from the denominator. This technique is widely used in various areas of mathematics, including algebra, trigonometry, and calculus. Understanding and applying this technique is essential for simplifying expressions and solving mathematical problems involving radicals.
Step 3: Perform the Multiplication
Now, we substitute the simplified square root back into our original expression:
30 * ((3√5) / 5)
We can rewrite this as:
(30 * 3√5) / 5
Multiply 30 by 3:
90√5 / 5
Now, divide 90 by 5:
90 ÷ 5 = 18
So, our final simplified expression is:
18√5
Performing the multiplication is the final step in simplifying the expression. This step involves combining the simplified square root with the remaining coefficient. By multiplying the coefficient outside the square root with the simplified radical, we obtain the final simplified form of the expression. This step often involves basic arithmetic operations such as multiplication and division, which are fundamental skills in mathematics. Ensuring accuracy in these operations is crucial to arriving at the correct final answer. The result, 18√5, is the simplest form of the original expression, as it cannot be further simplified. This final form is easier to understand and work with in further calculations or applications. Understanding how to perform this final multiplication is essential for completing the simplification process and arriving at a clear and concise solution.
Final Answer
Therefore, the simplified form of 30√(45/25) is 18√5.
Conclusion
In summary, to simplify the expression 30√(45/25), we followed a series of steps that involved simplifying fractions, simplifying square roots, and performing arithmetic operations. First, we simplified the fraction inside the square root, reducing 45/25 to 9/5. This step made the subsequent calculations easier by working with smaller numbers. Next, we simplified the square root √(9/5) by recognizing that it could be rewritten as √9 / √5. We then simplified √9 to 3 and rationalized the denominator by multiplying both the numerator and the denominator by √5, resulting in (3√5) / 5. Finally, we multiplied the simplified square root by 30, which gave us (30 * 3√5) / 5. After simplifying this expression, we arrived at the final answer of 18√5.
Each step in this simplification process is grounded in fundamental mathematical principles. Simplifying fractions involves finding the greatest common divisor and dividing both the numerator and the denominator by it. Simplifying square roots requires understanding how to break down numbers into their prime factors and identify perfect squares. Rationalizing the denominator is a technique used to eliminate square roots from the denominator, making the expression easier to work with. Arithmetic operations, such as multiplication and division, are used throughout the process to combine and simplify terms. By mastering these principles, we can confidently tackle a wide range of simplification problems.
This problem illustrates the importance of breaking down complex expressions into smaller, more manageable parts. By approaching the problem step by step, we can avoid errors and gain a deeper understanding of the underlying mathematical concepts. Simplifying expressions is a crucial skill in mathematics, as it allows us to solve more complex problems and make calculations more efficient. The final answer, 18√5, is the simplest form of the original expression, and it provides a clear and concise representation of the value. Understanding how to simplify expressions is not only essential for academic success in mathematics but also for practical applications in various fields, including science, engineering, and finance.
