5, Mar 2024
Is 2025 A Perfect Square?

Is 2025 a Perfect Square?

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Is 2025 a Perfect Square?

Show that each of the following number is a perfect square. In each case find the number whose

Introduction

A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 4^2. In this article, we will investigate whether the number 2025 is a perfect square.

Mathematical Analysis

To determine if a number is a perfect square, we can use the following steps:

  1. Find the prime factorization of the number.
  2. Pair the prime factors in groups of two.
  3. If all the prime factors can be paired, then the number is a perfect square.

Prime Factorization of 2025

The prime factorization of 2025 is:

2025 = 3 * 3 * 5 * 5 * 5

Pairing the Prime Factors

We can pair the prime factors as follows:

(3, 3)
(5, 5)
(5, 5)

Conclusion

Since all the prime factors can be paired, we can conclude that 2025 is a perfect square.

Finding the Square Root

To find the square root of 2025, we can use the following steps:

  1. Take the prime factorization and group the prime factors into pairs.
  2. For each pair of prime factors, take the square root of the product.
  3. Multiply the square roots together to get the square root of the original number.

Square Root of 2025

Using the prime factorization, we can find the square root of 2025 as follows:

โˆš2025 = โˆš(3 * 3 * 5 * 5 * 5)
= โˆš(3^2 * 5^3)
= 3 * 5โˆš5
= 15โˆš5

Properties of Perfect Squares

Perfect squares have several interesting properties:

  • The last digit of a perfect square is always 0, 1, 4, 5, 6, or 9.
  • The sum of the digits of a perfect square is always divisible by 3.
  • A number is a perfect square if and only if its binary representation has an even number of 1’s.

Applications of Perfect Squares

Perfect squares have many applications in mathematics and science, including:

  • Geometry: Calculating the area and volume of squares and cubes.
  • Algebra: Solving quadratic equations and factoring polynomials.
  • Trigonometry: Finding the values of trigonometric functions.
  • Physics: Calculating the energy levels of atoms and molecules.

Conclusion

The number 2025 is a perfect square. Its square root is 15โˆš5. Perfect squares have several interesting properties and are used in a wide variety of applications in mathematics and science.

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