Simplifying The Square Root Of 39 In Radical Form

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The concept of simplifying square roots is a fundamental idea in mathematics, particularly in algebra and geometry. Simplifying the square root of a number involves finding the largest perfect square that divides the number, and then expressing the result in a simplified radical form. In this article, we will focus on simplifying the square root of 39 in radical form.

The square root of 39 can be simplified by finding the largest perfect square that divides 39. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it can be expressed as 2 × 2, and 9 is a perfect square because it can be expressed as 3 × 3.

To simplify the square root of 39, we need to find the largest perfect square that divides 39. The perfect squares that divide 39 are 1 and 9. However, 9 is the largest perfect square that divides 39, so we will use 9 to simplify the square root.

The Simplification Process

The simplification process involves dividing the square root of 39 by the square root of 9, which is 3. This gives us:

√39 = √(9 × 4 + 3) = √(9 × (4 + 3/9)) = √(9 × (4 + 1/3)) = √9 × √(4 + 1/3) = 3√(4 + 1/3)

So, the simplified radical form of the square root of 39 is 3√(4 + 1/3) or √(39) = √(9*4+3) = 3√(4+3/9).

Rationalizing the Denominator

Rationalizing the denominator involves getting rid of any radicals in the denominator. In this case, we have a radical in the denominator, which is √(4 + 1/3). To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is √(4 - 1/3).

This gives us:

√(4 + 1/3) = (√(4 + 1/3) × √(4 - 1/3)) / (√(4 - 1/3) × √(4 - 1/3)) = (√(4 + 1/3) × √(4 - 1/3)) / (√(16 - 1/9)) = (√(4 + 1/3) × √(4 - 1/3)) / (√(144/9)) = (√(4 + 1/3) × √(4 - 1/3)) / (√(16)) = (√(4 + 1/3) × √(4 - 1/3)) / 4

Simplifying this expression, we get:

√(4 + 1/3) = (√(16 - 1/9) + √(16 + 1/9)) / 4

So, the simplified radical form of the square root of 39 is 3√(4 + 1/3) or 3(√(16 - 1/9) + √(16 + 1/9)) / 4.

Benefits of Simplifying Radicals

Simplifying radicals has several benefits, including:

• Easier calculations: Simplifying radicals makes it easier to perform calculations involving square roots. By expressing the square root in a simplified radical form, we can avoid having to deal with complicated expressions.
• Improved accuracy: Simplifying radicals can also improve accuracy in calculations. By expressing the square root in a simplified radical form, we can avoid errors that can occur when dealing with complicated expressions.
• Better understanding: Simplifying radicals can also help us better understand the properties of square roots. By expressing the square root in a simplified radical form, we can see how the square root is related to the original number.

Real-World Applications

Simplifying radicals has several real-world applications, including:

• Engineering: Simplifying radicals is used in engineering to calculate distances and angles. For example, when building a bridge, engineers need to calculate the length of the bridge's supports, which involves simplifying radicals.
• Physics: Simplifying radicals is used in physics to calculate velocities and accelerations. For example, when calculating the velocity of an object, physicists need to simplify radicals to get an accurate answer.
• Mathematics: Simplifying radicals is used in mathematics to solve equations and inequalities. For example, when solving a quadratic equation, mathematicians need to simplify radicals to get an accurate answer.

Conclusion

In conclusion, simplifying the square root of 39 in radical form involves finding the largest perfect square that divides 39 and then expressing the result in a simplified radical form. The simplified radical form of the square root of 39 is 3√(4 + 1/3) or 3(√(16 - 1/9) + √(16 + 1/9)) / 4. Simplifying radicals has several benefits, including easier calculations, improved accuracy, and better understanding. It also has several real-world applications, including engineering, physics, and mathematics.

We hope this article has helped you understand how to simplify the square root of 39 in radical form. If you have any questions or need further clarification, please don't hesitate to ask. We're always here to help.