What's the Square Root of 90? Discover the Answer and More

The square root of 90 can be expressed in various forms including its decimal form, radical form, and exponential form.

Decimal Form

The square root of 90 in decimal form is approximately:

\(\sqrt{90} \approx 9.486832980505138\)

Radical Form

In radical form, the square root of 90 is expressed as:

\(\sqrt{90} = 3\sqrt{10}\)

Exponential Form

In exponential form, the square root of 90 is written as:

\(90^{1/2}\) or \(90^{0.5}\)

Estimation Method

By recognizing that \(9^2 = 81\) and \(10^2 = 100\), we can estimate that \(\sqrt{90}\) lies between 9 and 10. Fine-tuning this estimate leads to the value of approximately 9.49.

Long Division Method

This method involves a step-by-step manual calculation:

1. Write 90 as 90.00 00 00, breaking it into pairs of digits after the decimal point.
2. Find a number whose square is closest to 90 without exceeding it (9 in this case).
3. Subtract 81 from 90 to get a remainder of 9.
4. Bring down the next pair of zeros, making the new dividend 900.
5. Double the initial quotient (9) to get 18, and use it as the new divisor (180).
6. Find a digit to add to 180 to form a new divisor (184) such that 184×4=736, which is as close as possible to 900 without exceeding it.
7. Subtract 736 from 900 to get 164.
8. Repeat the process until the desired precision is achieved.

• Rational or Irrational: The square root of 90 is an irrational number because it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating.
• Perfect Square: 90 is not a perfect square as its square root is not an integer.
• Real Number: Yes, the square root of 90 is a real number.

Between Which Two Integers Does the Square Root of 90 Lie?

The square root of 90 lies between 9 and 10.

How is the Square Root of 90 Expressed in Different Forms?

In radical form: \(3\sqrt{10}\)

In exponential form: \(90^{1/2}\)

What is the Square Root of 90 Rounded to the Nearest Tenth?

The square root of 90 rounded to the nearest tenth is 9.5.

For more information, you can visit resources such as Cuemath, Mathway, Symbolab, Square Root Calculator, Examples, and CoolConversion.

Estimation Method

By recognizing that \(9^2 = 81\) and \(10^2 = 100\), we can estimate that \(\sqrt{90}\) lies between 9 and 10. Fine-tuning this estimate leads to the value of approximately 9.49.

Long Division Method

This method involves a step-by-step manual calculation:

1. Write 90 as 90.00 00 00, breaking it into pairs of digits after the decimal point.
2. Find a number whose square is closest to 90 without exceeding it (9 in this case).
3. Subtract 81 from 90 to get a remainder of 9.
4. Bring down the next pair of zeros, making the new dividend 900.
5. Double the initial quotient (9) to get 18, and use it as the new divisor (180).
6. Find a digit to add to 180 to form a new divisor (184) such that 184×4=736, which is as close as possible to 900 without exceeding it.
7. Subtract 736 from 900 to get 164.
8. Repeat the process until the desired precision is achieved.

• Rational or Irrational: The square root of 90 is an irrational number because it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating.
• Perfect Square: 90 is not a perfect square as its square root is not an integer.
• Real Number: Yes, the square root of 90 is a real number.

Between Which Two Integers Does the Square Root of 90 Lie?

The square root of 90 lies between 9 and 10.

How is the Square Root of 90 Expressed in Different Forms?

In radical form: \(3\sqrt{10}\)

In exponential form: \(90^{1/2}\)

What is the Square Root of 90 Rounded to the Nearest Tenth?

The square root of 90 rounded to the nearest tenth is 9.5.

For more information, you can visit resources such as Cuemath, Mathway, Symbolab, Square Root Calculator, Examples, and CoolConversion.

• Rational or Irrational: The square root of 90 is an irrational number because it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating.
• Perfect Square: 90 is not a perfect square as its square root is not an integer.
• Real Number: Yes, the square root of 90 is a real number.

Between Which Two Integers Does the Square Root of 90 Lie?

The square root of 90 lies between 9 and 10.

How is the Square Root of 90 Expressed in Different Forms?

In radical form: \(3\sqrt{10}\)

In exponential form: \(90^{1/2}\)

What is the Square Root of 90 Rounded to the Nearest Tenth?

The square root of 90 rounded to the nearest tenth is 9.5.

For more information, you can visit resources such as Cuemath, Mathway, Symbolab, Square Root Calculator, Examples, and CoolConversion.

Between Which Two Integers Does the Square Root of 90 Lie?

The square root of 90 lies between 9 and 10.

How is the Square Root of 90 Expressed in Different Forms?

In radical form: \(3\sqrt{10}\)

In exponential form: \(90^{1/2}\)

What is the Square Root of 90 Rounded to the Nearest Tenth?

The square root of 90 rounded to the nearest tenth is 9.5.

For more information, you can visit resources such as Cuemath, Mathway, Symbolab, Square Root Calculator, Examples, and CoolConversion.

The square root of 90 is a fundamental mathematical concept with various practical applications. It is an irrational number, which means it cannot be expressed as a simple fraction and its decimal form is non-repeating and non-terminating. Understanding how to calculate and simplify the square root of 90 can be useful in fields such as geometry, algebra, and real-world problem solving. Let's explore the different methods to find and simplify the square root of 90 step by step.

• √90 in radical form is expressed as 3√10.
• √90 approximately equals 9.486832980505138.

Various methods to calculate the square root of 90 include estimation, using a calculator, and the long division method. Each method provides different levels of precision and is suitable for different contexts.

The principal square root of a number is its positive square root. For any positive number \( x \), the principal square root is denoted as \( \sqrt{x} \). Every positive number has two square roots: one positive and one negative. However, the principal square root is always the positive one.

For example, the principal square root of 90 is the positive number that, when multiplied by itself, equals 90. This can be expressed mathematically as:

\[ \sqrt{90} \approx 9.4868329805051 \]

This value means that \( 9.4868329805051 \times 9.4868329805051 = 90 \).

To calculate the principal square root of 90, you can use various methods:

• Using a calculator: Simply enter 90 and press the square root button (\(\sqrt{x}\)).
• Using Excel or Google Sheets: Type =SQRT(90) in a cell and press enter.

The principal square root is widely used in various mathematical contexts and applications, especially in solving equations and analyzing geometric properties.

In conclusion, the principal square root of 90 is approximately 9.4868329805051, which is often rounded to 9.487 for simplicity.

The square root of 90 can be simplified by breaking it down into its prime factors and applying the properties of radicals. Here is a step-by-step method:

1. Factorize 90 into its prime factors:
   • 90 = 9 × 10
   • 9 = 3 × 3
   • 10 = 2 × 5
   • Therefore, 90 = 3 × 3 × 2 × 5
2. Group the prime factors into pairs:
   • 90 = (3 × 3) × (2 × 5)
3. Apply the square root to the groups:
   • √90 = √((3 × 3) × (2 × 5))
   • √90 = √(3² × 10)
4. Simplify the square root:
   • √90 = 3√10
5. Convert to decimal form if needed:
   • 3√10 ≈ 3 × 3.162 ≈ 9.4868

Thus, the simplified form of the square root of 90 is 3√10, which is approximately 9.4868.

The square root of a number has several important properties that are useful in various mathematical contexts. Understanding these properties can help simplify complex calculations and solve problems more efficiently.

• Exponential Form: The square root of a number \(x\) can be expressed as \(x^{1/2}\) or \(x^{0.5}\). For example, the square root of 90 can be written as \(90^{1/2}\).
• Integer Results: If the square root of a number is an integer, the original number must end in 0, 1, 4, 5, 6, or 9. For example, \( \sqrt{100} = 10 \) and \( \sqrt{81} = 9 \).
• Even and Odd Numbers: The square root of an even number is always even, and the square root of an odd number is always odd. For example, \( \sqrt{64} = 8 \) (both even) and \( \sqrt{121} = 11 \) (both odd).
• Last Digit Rule: If a number ends in 2, 3, 7, or 8, its square root is not an integer. This helps in quickly determining whether a number has an integer square root.
• Positive and Negative Roots: For any positive number \(X\), both \( \sqrt{X} \) and \( -\sqrt{X} \) are considered square roots. This is often expressed as \( \sqrt{X} = \pm x \).
• Product Rule: The square root of a product is the product of the square roots. For two numbers \(x\) and \(y\), \( \sqrt{x \times y} = \sqrt{x} \times \sqrt{y} \). This property simplifies the computation of square roots of large products.
• Sum of Odd Numbers: The square of a natural number \(n\) is the sum of the first \(n\) odd numbers. For example, \( 5^2 = 25 \) is the sum of the first five odd numbers: 1, 3, 5, 7, and 9.
• Complex Numbers: The square root of a negative number does not exist within the set of real numbers but does in the set of complex numbers. For example, \( \sqrt{-7} = 7i \), where \( i = \sqrt{-1} \).
• Rationality: The square root of 90 is an irrational number because it cannot be expressed as a simple fraction. Its decimal expansion is non-repeating and non-terminating.

Understanding the square root of 90 provides valuable insight into various mathematical applications. The principal square root of 90, approximately 9.4868329805, highlights its non-perfect square nature and its representation as an irrational number. This value is essential for solving problems in algebra, geometry, and other mathematical fields where precise measurements and calculations are required.

The square root of 90 can be computed using several methods, each offering different levels of precision and understanding. Estimation allows for quick approximations, recognizing that √90 lies between 9 and 10. Using a calculator provides an accurate and immediate result, while the long division method and the Babylonian method offer more detailed and step-by-step calculations, enhancing our understanding of the process.

Additionally, the properties of the square root of 90, including its representation in radical form (3√10) and its principal and negative values (±9.4868329805), underscore its role in various mathematical contexts. These properties are crucial when simplifying expressions and solving equations involving square roots.

Recognizing the significance of the square root of 90, we can appreciate its applications in real-world scenarios. For example, determining the side length of a square garden with an area of 90 square yards involves calculating √90, which is approximately 9.49 yards. This practical understanding extends to various disciplines, making the knowledge of square roots an essential tool in mathematics and beyond.