Square & Square Root of 87

Within algebraic mathematics, squares and square roots are fundamental concepts. Squaring a number, as seen with 87 (87² = 7569), involves multiplying the number by itself and is crucial for exploring rational and irrational numbers. Rational numbers can be expressed as fractions of integers, while irrational numbers cannot. Understanding these basics enriches mathematical comprehension and reveals intricate relationships and patterns.

Square of 87

87² (87 × 87) = 7569

The square number of 87, a fundamental mathematical concept, yields 7569. This operation of multiplying a number by itself is essential for algebraic exploration and understanding properties of numbers, both rational and irrational. It helps reveal mathematical patterns and relationships.

Square Root of 87

√87 = 9.32737905309

The square root of 87, another fundamental mathematical concept, is approximately 9.32737905309. This value represents the number which, when multiplied by itself, equals 87. Mastery of square roots is crucial for understanding algebraic relationships and properties of both rational and irrational numbers, facilitating problem-solving across diverse domains.

Square Root of 87: 9.32737905309

Exponential Form: 87^½ or 87^0.5

Radical Form: √87

Is the Square Root of 87 Rational or Irrational?

The square root of 87 is an irrational number.

Irrational numbers cannot be expressed as a fraction of two integers and their decimal representations neither terminate nor repeat. Thus, √87 cannot be expressed as a simple fraction, and its decimal representation continues infinitely without repeating.

A rational number is any number that can be expressed as the fraction of two integers, where the denominator is not zero..

Methods to Find the Value of √87

• Prime Factorization Method: Since 87 is a product of prime factors (3, 29), and none form pairs, this method confirms the irrationality without providing an exact value.
• Estimation Method: Approximate √87 by finding perfect squares close to it. Since 87 is between 81 (9²) and 100 (10²), its square root is between 9 and 10. Refine the estimate using methods like Newton’s method.
• Long Division Method: This method finds the square root iteratively by dividing and averaging. It’s reliable for manual calculations.
• Calculator or Software: Use a calculator or software to directly compute √87. These tools have built-in functions for square roots.
• Continued Fraction Expansion: This method expresses √87 as a continued fraction and approximates its value by truncating the expansion.

Square Root of 87 by Long Division Method

Calculating Square Root of 87 by Long Division Method

• Initial Setup:

Pair the digits of 87 (87.000000) and find the largest number whose square is ≤ 87.

9 × 9 = 81. Place 9 in the quotient and subtract 81 from 87 to get 6.

• Form New Divisor:

Double the quotient (9) to get 18.

Bring down the next pair of zeros, making the new dividend 600.

183 × 3 = 549. Place 3 in the quotient.

• Continue Division:

Subtract 549 from 600 to get 51.

Bring down the next pair of zeros, making the new dividend 5100.

1862 × 2 = 3724. Place 2 in the quotient.

• Repeat Steps:

Subtract 3724 from 5100 to get 1376.

Bring down the next pair of zeros, making the new dividend 137600.

18647 × 7 = 130529. Place 7 in the quotient.

• Finalize:

Subtract 130529 from 137600 to get 7071.

Quotient now reads 9.327.

Hence, √87 ≈ 9.327.

Is 87 a Perfect Square?

No, 87 is not a perfect square.

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 9 is a perfect square because it equals 3 × 3. However, 87 cannot be expressed as the product of two equal integers, so it is not a perfect square.