Simplifying 2√2/(√3-√2) A Step By Step Solution
In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent complex mathematical statements in a more concise and understandable manner. This article delves into the process of simplifying the expression 2√2/(√3-√2), providing a step-by-step guide and explaining the underlying mathematical principles.
Understanding the Problem: Simplifying Radical Expressions
Before we dive into the solution, it's crucial to grasp the core concept: simplifying radical expressions. Radical expressions involve roots, such as square roots (√), cube roots, and so on. Simplifying these expressions often entails removing radicals from the denominator, a process known as rationalizing the denominator. This makes the expression easier to work with and compare to other expressions. The given expression, 2√2/(√3-√2), presents a classic example where rationalizing the denominator is necessary.
Our primary goal in simplifying radical expressions is to eliminate any radicals from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the terms in the denominator. For instance, the conjugate of (√3 - √2) is (√3 + √2). This technique leverages the difference of squares identity, which states that (a - b)(a + b) = a² - b². By multiplying the denominator by its conjugate, we effectively square the radical terms, eliminating the square roots from the denominator. This process transforms the expression into a more manageable form, making it easier to perform further calculations or comparisons. Rationalizing the denominator is a key skill in simplifying expressions and is frequently used in various mathematical contexts.
The given expression, 2√2/(√3-√2), includes a radical expression in the denominator, which is (√3-√2). To simplify this, we will employ the technique of rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (√3-√2) is (√3+√2). By doing so, we aim to eliminate the radical from the denominator, making the expression simpler and easier to work with. This method is widely used in algebra and calculus to simplify expressions involving radicals and fractions. The ability to rationalize denominators is a fundamental skill that allows us to manipulate and understand mathematical expressions more effectively. In the following sections, we will demonstrate the step-by-step process of rationalizing the denominator and simplifying the given expression.
Step-by-Step Solution: Rationalizing the Denominator
To simplify the expression 2√2/(√3-√2), we begin by rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator, which is (√3 + √2).
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Multiply by the Conjugate:
Multiply both the numerator and denominator by (√3 + √2):
(2√2 / (√3 - √2)) * ((√3 + √2) / (√3 + √2))
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Expand the Numerator:
Multiply 2√2 by (√3 + √2):
2√2 * (√3 + √2) = 2√2 * √3 + 2√2 * √2 = 2√6 + 2 * 2 = 2√6 + 4
When expanding the numerator, we distribute 2√2 across both terms within the parentheses. First, we multiply 2√2 by √3, resulting in 2√6. Then, we multiply 2√2 by √2. Since √2 * √2 equals 2, the second term becomes 2 * 2, which simplifies to 4. Combining these results, we get 2√6 + 4. This step is crucial in simplifying the expression as it eliminates the need to divide by a radical term. The expanded numerator, 2√6 + 4, will now be divided by the simplified denominator, which we will calculate in the next step. This process is a key component of rationalizing denominators and simplifying radical expressions in mathematics.
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Expand the Denominator:
Multiply (√3 - √2) by (√3 + √2). This is in the form of (a - b)(a + b) = a² - b²:
(√3 - √2)(√3 + √2) = (√3)² - (√2)² = 3 - 2 = 1
When expanding the denominator, we utilize the difference of squares identity, which states that (a - b)(a + b) = a² - b². In our case, a is √3 and b is √2. Applying this identity, we square both terms: (√3)² and (√2)². The square of √3 is 3, and the square of √2 is 2. Subtracting these values, we get 3 - 2, which equals 1. This step is the heart of rationalizing the denominator, as it eliminates the radicals from the denominator, simplifying it to a rational number. A denominator of 1 greatly simplifies the expression, making it easier to understand and work with. This technique is widely used in algebra and calculus to manipulate and simplify radical expressions.
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Simplify the Expression:
Now we have: (2√6 + 4) / 1
Since dividing by 1 doesn't change the value, the simplified expression is:
2√6 + 4
Final Answer: The Simplified Form
Therefore, the simplest form of the expression 2√2/(√3-√2) is 2√6 + 4. This result is obtained by rationalizing the denominator, which involves multiplying both the numerator and the denominator by the conjugate of the denominator. This process eliminates the radical from the denominator, leading to a simplified expression. The simplified form, 2√6 + 4, is easier to understand and work with compared to the original expression.
The simplified form, 2√6 + 4, represents the same value as the original expression but is presented in a more concise and manageable format. This simplification not only makes the expression easier to interpret but also facilitates further mathematical operations. For example, if we needed to compare this value with another expression or use it in a calculation, the simplified form would be significantly more convenient. The ability to simplify expressions is a fundamental skill in mathematics, allowing us to manipulate and understand complex mathematical relationships more effectively.
In conclusion, the process of simplifying radical expressions, such as 2√2/(√3-√2), is a vital skill in mathematics. By rationalizing the denominator, we eliminate radicals from the denominator, making the expression easier to work with. The final simplified form, 2√6 + 4, showcases the power of algebraic manipulation in achieving clarity and conciseness in mathematical expressions.
