Simplifying Radical Expressions And Representing √56 On The Number Line

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In mathematics, simplifying expressions involving radicals is a crucial skill. Radicals, often represented by the square root symbol (√), can sometimes appear in complex fractions or expressions. To effectively work with these expressions, it's essential to know how to rationalize denominators and simplify radicals. In this article, we will delve into the step-by-step process of simplifying two such expressions:

i) (6 - 4√2) / (6 + 4√2)

ii) 30 / (5√3 - 3√5)

i) Simplifying (6 - 4√2) / (6 + 4√2)

The primary goal here is to eliminate the radical from the denominator. This process is known as rationalizing the denominator. To achieve this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (6 + 4√2) is (6 - 4√2). This technique leverages the difference of squares identity, which states that (a + b)(a - b) = a² - b². By multiplying by the conjugate, we transform the denominator into a rational number.

Let's break down the steps:

  1. Identify the Expression: The given expression is (6 - 4√2) / (6 + 4√2).

  2. Find the Conjugate: The conjugate of the denominator (6 + 4√2) is (6 - 4√2).

  3. Multiply by the Conjugate: Multiply both the numerator and the denominator by (6 - 4√2):

    [(6 - 4√2) / (6 + 4√2)] * [(6 - 4√2) / (6 - 4√2)]

  4. Expand the Numerator: Multiply the numerators:

    (6 - 4√2) * (6 - 4√2) = 6 * 6 - 6 * 4√2 - 4√2 * 6 + 4√2 * 4√2

    = 36 - 24√2 - 24√2 + 16 * 2

    = 36 - 48√2 + 32

    = 68 - 48√2

  5. Expand the Denominator: Multiply the denominators:

    (6 + 4√2) * (6 - 4√2) = 6 * 6 - 6 * 4√2 + 4√2 * 6 - 4√2 * 4√2

    = 36 - 24√2 + 24√2 - 16 * 2

    = 36 - 32

    = 4

  6. Simplify the Expression: Now, the expression looks like this:

    (68 - 48√2) / 4

    Divide both terms in the numerator by the denominator:

    = 68/4 - (48√2)/4

    = 17 - 12√2

Therefore, the simplified form of (6 - 4√2) / (6 + 4√2) is 17 - 12√2. This process of rationalizing the denominator has allowed us to express the fraction in a more manageable form, devoid of radicals in the denominator. Understanding and applying the concept of conjugates is key to mastering this technique.

ii) Simplifying 30 / (5√3 - 3√5)

Similar to the previous example, we aim to rationalize the denominator of the expression 30 / (5√3 - 3√5). This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of (5√3 - 3√5) is (5√3 + 3√5). This method will help us eliminate the radicals from the denominator, making the expression simpler.

Here’s a step-by-step simplification:

  1. Identify the Expression: The given expression is 30 / (5√3 - 3√5).

  2. Find the Conjugate: The conjugate of the denominator (5√3 - 3√5) is (5√3 + 3√5).

  3. Multiply by the Conjugate: Multiply both the numerator and the denominator by (5√3 + 3√5):

    [30 / (5√3 - 3√5)] * [(5√3 + 3√5) / (5√3 + 3√5)]

  4. Expand the Numerator: Multiply the numerator:

    30 * (5√3 + 3√5) = 30 * 5√3 + 30 * 3√5

    = 150√3 + 90√5

  5. Expand the Denominator: Multiply the denominators:

    (5√3 - 3√5) * (5√3 + 3√5) = (5√3 * 5√3) + (5√3 * 3√5) - (3√5 * 5√3) - (3√5 * 3√5)

    = 25 * 3 + 15√15 - 15√15 - 9 * 5

    = 75 - 45

    = 30

  6. Simplify the Expression: Now, the expression looks like this:

    (150√3 + 90√5) / 30

    Divide both terms in the numerator by the denominator:

    = (150√3) / 30 + (90√5) / 30

    = 5√3 + 3√5

Thus, the simplified form of 30 / (5√3 - 3√5) is 5√3 + 3√5. This simplification not only removes the radicals from the denominator but also presents the expression in a clearer, more concise format. Rationalizing the denominator is a valuable technique for simplifying complex expressions and is frequently used in various mathematical contexts.

Representing irrational numbers such as √56 on the number line requires a geometric approach. The method involves utilizing the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a² + b² = c², where c is the hypotenuse, and a and b are the other two sides.

To represent √56, we need to find two integers, a and b, such that a² + b² = 56. This will allow us to construct a right-angled triangle with sides of length a and b, where the hypotenuse will have a length of √56. Finding these integers and accurately constructing the triangle are crucial steps in this process.

Let's break down the process into detailed steps:

  1. Express 56 as a Sum of Squares: We need to find two perfect squares that add up to 56. By trying different combinations, we find that 56 can be expressed as 49 + 7, which is 7² + (√7)². However, using √7 directly is complex for a number line representation. A more straightforward approach is to consider 56 as 4 + 52, which translates to 2² + (√52)². Now, we need to further break down √52. √52 can be written as √(4 + 48), which is √(2² + (√48)²). This decomposition can continue, but it leads to complex structures. A more direct approach is recognizing that 56 is between 49 (7²) and 64 (8²). Let’s try expressing 56 as a combination involving smaller perfect squares. We can express 56 as 4 * 14. However, this doesn’t directly translate into a sum of two perfect squares. Instead, let’s consider the nearest perfect square less than 56, which is 49 (7²). Then, 56 - 49 = 7. So, we have √56 ≈ √49 + √7, but this is an approximation and not suitable for geometric representation.

    Another approach is to express 56 as a product of its prime factors: 56 = 2 * 2 * 2 * 7. This can be rearranged as 56 = 4 * 14 = 2² * 14. This doesn't immediately give us two perfect squares, but it suggests a different method.

    Let's try a different factorization approach. We can write 56 as 4 * 14. Now, we express 14 as a sum of squares. We can write 14 as 9 + 5, which is 3² + (√5)². However, this involves another radical. A more direct decomposition is 56 = 49 + 7 = 7² + (√7)². But this doesn’t give us integers.

    The key insight here is to use a different strategy. Instead of directly finding two perfect squares that sum up to 56, we will use a spiral method on the number line.

  2. Draw the Number Line: Start by drawing a number line and mark the integer points. Label the point 0 as O. This line serves as the foundation for our representation, providing a visual context for the placement of √56. The accuracy of the final representation depends on the precision of this initial step. Using a ruler and ensuring equal spacing between integers will enhance the accuracy of the representation.

  3. Locate √49 (7): Since √49 is 7, mark the point 7 on the number line. This serves as our starting point for building up to √56. Marking this integer clearly is essential for the subsequent steps, as it provides a reference for constructing the right-angled triangle. Ensuring this point is accurately marked is critical for the rest of the construction.

  4. Construct a Perpendicular Line: At the point 7, construct a line perpendicular to the number line. This perpendicular line forms one of the legs of the right-angled triangle we are about to construct. Using a protractor or a set square will ensure the perpendicular line is accurately drawn. The precision of this perpendicular line will directly impact the accuracy of the final representation.

  5. Mark a Length of √7 on the Perpendicular Line: We need a length such that when squared and added to 7², it gives us 56. Since 56 = 49 + 7, we need a length of √7 on the perpendicular line. To construct √7, we can use the spiral method. First, mark a length of 1 unit on the number line from 7, reaching 8. Then, draw a perpendicular line of 1 unit at 8. The hypotenuse will be √2. Next, at the end of √2, draw a perpendicular of 1 unit; the hypotenuse will be √3. Continue this process until you reach √7. The easier approach for this problem is √7 as the perpendicular to the 7 point.

  6. Draw the Hypotenuse: Draw a line connecting the origin (0) to the endpoint of √7 on the perpendicular line. This line represents the hypotenuse of the right-angled triangle. According to the Pythagorean theorem, the length of this hypotenuse is √(7² + (√7)²) = √(49 + 7) = √56. This hypotenuse is the key to representing √56 on the number line.

  7. Transfer the Length to the Number Line: Using a compass, place the center at the origin (0) and the pencil at the endpoint of the hypotenuse. Draw an arc that intersects the number line. The point of intersection represents √56 on the number line. This step effectively transfers the length of the hypotenuse onto the number line, allowing us to visualize the position of √56. The accuracy of the compass and the precision in drawing the arc are crucial for accurately locating √56.

By following these steps, we accurately represent √56 on the number line. This method combines geometric construction with the principles of the Pythagorean theorem to visualize irrational numbers, providing a tangible understanding of their position within the real number system. This representation not only helps in visualizing the number but also reinforces the relationship between algebra and geometry.

Simplifying expressions with radicals and representing irrational numbers on the number line are fundamental skills in mathematics. Through the techniques of rationalizing denominators and geometric constructions, we can manipulate and visualize these numbers more effectively. Mastering these concepts enhances our ability to work with mathematical expressions and understand the nature of numbers.

By understanding the properties of conjugates, applying the Pythagorean theorem, and employing careful geometric constructions, we can confidently tackle problems involving radicals and irrational numbers. These skills are not only essential for academic success but also provide a solid foundation for advanced mathematical studies.