What is the Square Root of 90: Understanding and Calculation

The square root of 90 can be expressed in several ways, including its exact form, simplified radical form, and decimal form. Here's a detailed explanation:

Exact Form

The square root of 90 is given by:

\(\sqrt{90} = 9.486832980505138\)

Simplified Radical Form

The square root of 90 can be simplified by factoring out the largest perfect square:

\(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\)

Decimal Form

In decimal form, the square root of 90 is approximately:

\(\sqrt{90} \approx 9.4868\)

How to Calculate

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

• Using a calculator, simply enter 90 and press the square root button.
• In Excel or Google Sheets, use the formula =SQRT(90) to get the result.
• By hand, you can use the long division method to approximate the square root to several decimal places.

Examples

1. Example 1: Simplify \(\sqrt{90}\)

   Simplify using the product rule for radicals:

2. Example 2: Calculate \(\sqrt{90}\) in decimal form

   Using a calculator, \(\sqrt{90} \approx 9.486832980505138\)

3. Example 3: Fence a square garden with an area of 90 sq yards

   Given area = 90 sq yards

   Side of the garden = \(\sqrt{90} \approx 9.48\) yards

   Perimeter = \(4 \times 9.48 \approx 37.92\) yards

FAQs

• What is the square root of 90 simplified? The simplified form is \(3\sqrt{10}\).
• Is the square root of 90 a rational number? No, it is an irrational number.
• Between which two integers does \(\sqrt{90}\) lie? It lies between 9 and 10.

The square root of 90 is an interesting mathematical concept with various applications and properties. It is represented as √90 in radical form, and its approximate value is 9.4868. In this section, we will explore what a square root is, why the square root of 90 is irrational, and how to calculate it using different methods.

• The square root of 90 is not a perfect square because it cannot be expressed as the product of an integer with itself.
• It is an irrational number, meaning it cannot be exactly written as a fraction of two integers, and its decimal representation is non-repeating and non-terminating.
• One way to calculate the square root of 90 is by using a calculator, which gives a precise value of approximately 9.486832980505.
• Another method is the long division technique, which involves step-by-step manual calculations to achieve a high level of precision.
• We can also approximate the square root through estimation by recognizing that it lies between the squares of 9 and 10 (since 9²=81 and 10²=100).

Understanding the square root of 90 provides a foundation for learning about more complex mathematical concepts and enhances problem-solving skills. In the following sections, we will delve deeper into these methods and their applications.

The square root of a number is a value that, when multiplied by itself, gives the original number. In other words, if y is the square root of x, then y × y = x. The square root is denoted by the radical symbol √. For example, the square root of 90 is represented as √90 or √90.

Mathematically, the square root function can be defined as:

y = √x where y is the square root of x.

Some key properties of square roots include:

• The square root of a positive number is always positive. For example, √90 is approximately 9.487.
• The square root of zero is zero: √0 = 0.
• There is no real square root of a negative number, since the product of two positive or two negative numbers is positive.

Square roots are often used in various mathematical contexts such as solving quadratic equations, in geometry to find the length of sides of right-angled triangles (Pythagorean theorem), and in many real-life applications where distance or magnitude is involved.

The square root of a number is a value that, when multiplied by itself, gives the original number. For the number 90, the mathematical representation of its square root involves understanding its radical and exponential forms.

The square root of 90 can be represented in the following ways:

• Radical Form: \( \sqrt{90} \)
• Exponential Form: \( 90^{1/2} \)

To simplify the square root of 90, we can use the prime factorization method:

1. First, factor 90 into its prime factors:
   • \( 90 = 2 \times 3^2 \times 5 \)
2. Next, express the square root using these factors:
   • \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \)
3. Simplify by taking the square root of the perfect square:
   • \( \sqrt{90} = \sqrt{3^2 \times 10} = 3\sqrt{10} \)

Thus, the simplified form of the square root of 90 is \( 3\sqrt{10} \).

In decimal form, the square root of 90 is approximately:

• \( \sqrt{90} \approx 9.4868 \)

This value is irrational, meaning it cannot be expressed as a simple fraction and has an infinite number of non-repeating decimals.

There are several methods to calculate the square root of 90. Here are the most common techniques:

1. Long Division Method

1. Write the number 90 in pairs of digits from right to left: 90.00 00 00.
2. Find the largest number whose square is less than or equal to 90. In this case, it's 9 (since 9^2 = 81).
3. Subtract 81 from 90, giving a remainder of 9. Bring down the next pair of zeros, making it 900.
4. Double the quotient obtained (9), which gives 18. Use this as the starting part of the new divisor.
5. Find a number to complete the divisor (18X), such that 18X * X is less than or equal to 900. Here, X is 4 (184 * 4 = 736).
6. Subtract 736 from 900 to get a remainder of 164. Bring down the next pair of zeros, making it 16400.
7. Repeat the process until you achieve the desired precision. The quotient obtained will be approximately 9.486.

2. Approximation Method

1. Start with an initial guess close to the square root of 90, for instance, 9.
2. Improve the guess using the formula: \( \text{new guess} = \frac{\text{old guess} + \frac{90}{\text{old guess}}}{2} \).
3. Continue this process until the guesses converge to a stable value. For example:
   • Initial guess: 9
   • Next guess: \( \frac{9 + \frac{90}{9}}{2} = 9.5 \)
   • Further guesses will refine to approximately 9.486.

3. Calculator Method

Most modern calculators have a square root function. Simply enter 90 and press the square root button (√) to get an approximate result: √90 ≈ 9.486.

4. Using Spreadsheets

You can also use spreadsheet software like Excel, Google Sheets, or Numbers:

• In any cell, type the formula =SQRT(90).
• The result will be approximately 9.486832980505.

To simplify the square root of 90, we need to express it in its simplest radical form. This involves breaking down the number inside the square root to its prime factors and extracting any perfect squares.

1. Factorize the Number:

   First, list the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Identify the perfect squares among these factors, which are 1 and 9.

2. Find the Largest Perfect Square:

   From the list of factors, 9 is the largest perfect square that can divide 90. So, we write 90 as a product of 9 and another number.

   \[ 90 = 9 \times 10 \]

3. Rewrite Using Square Root Properties:

   Use the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

   \[ \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} \]

4. Extract the Square Root of the Perfect Square:

   The square root of 9 is 3. Therefore, we can simplify \(\sqrt{90}\) as:

   \[ \sqrt{90} = 3\sqrt{10} \]

Thus, the simplified form of \(\sqrt{90}\) is \(3\sqrt{10}\). This means that 90 can be expressed in terms of a simpler radical expression, making calculations easier and more manageable.

In mathematics, numbers are classified into various categories based on their properties. Two of the main categories are rational and irrational numbers.

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where p and q are integers and q is not zero. In other words, a rational number can be written in the form of a fraction where both the numerator and the denominator are integers.

• Examples of rational numbers include:
  • \( \frac{1}{2} \)
  • \( -\frac{3}{4} \)
  • 5 (which can be written as \( \frac{5}{1} \))
  • 0.75 (which can be written as \( \frac{3}{4} \))
• Rational numbers include integers, finite decimals, and repeating decimals.

Irrational Numbers

An irrational number, on the other hand, cannot be expressed as a simple fraction. Irrational numbers have decimal expansions that neither terminate nor become periodic. This means their decimal form goes on forever without repeating.

• Examples of irrational numbers include:
  • π (pi), which is approximately 3.14159...
  • √2, which is approximately 1.41421...
  • e (Euler's number), which is approximately 2.71828...
• Irrational numbers are non-repeating and non-terminating decimals.

Key Differences Between Rational and Irrational Numbers

Understanding the distinction between rational and irrational numbers is fundamental in various areas of mathematics, including algebra, calculus, and number theory.

The process of simplifying the square root of 90 involves breaking down the number into its prime factors and identifying the perfect square factors. Here are the steps to achieve the simplified form:

1. List Factors:

   Identify the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

2. Identify Perfect Squares:

   From the list of factors, identify the perfect squares. The perfect squares are: 1, 9.

3. Divide by the Largest Perfect Square:

   Divide 90 by the largest perfect square, which is 9.

   \[
   \frac{90}{9} = 10
   \]

4. Square Root of the Perfect Square:

   Calculate the square root of 9.

   \[
   \sqrt{9} = 3
   \]

5. Combine Results:

   Combine the results to get the simplified form of the square root of 90.

   \[
   \sqrt{90} = 3 \sqrt{10}
   \]

In summary, the square root of 90 simplified in radical form is \( 3\sqrt{10} \). This means that while the exact value of the square root of 90 is approximately 9.486, its simplest radical form expresses it as a product of the square root of a perfect square and the square root of a remaining factor.

Additionally, understanding perfect squares is crucial in various mathematical contexts. Perfect squares are numbers that can be expressed as the product of an integer with itself, such as 1, 4, 9, 16, and so on. Simplifying radicals often involves identifying and extracting these perfect squares from the original number under the square root.

The square root of 90 has several interesting applications in both real-life situations and mathematical problems. Below are some examples and applications:

Real-life Examples

• Construction and Design: When determining dimensions for projects such as garden layouts or flooring, the square root of 90 can be used to find side lengths of squares with an area of 90 square units. For instance, if you need to create a square patio with an area of 90 square feet, each side of the patio would be approximately \( \sqrt{90} \approx 9.49 \) feet.
• Physics: The square root function is crucial in calculating time for objects in free fall. For example, the time it takes for an object to fall from a height can be calculated using the formula \( t = \frac{\sqrt{h}}{4} \). If the height (h) is 90 feet, then \( t = \frac{\sqrt{90}}{4} \approx 2.37 \) seconds.
• Accident Investigations: In forensic science, the speed of a vehicle can be estimated by measuring skid marks. The formula \( v = \sqrt{24d} \) uses the square root to find speed (v) from skid distance (d). For example, if skid marks are 90 feet long, the speed is \( v = \sqrt{24 \times 90} \approx 46.48 \) mph.

Mathematical Problems

• Finding Side Lengths: Given the area of a square is 90 square units, the side length of the square is \( \sqrt{90} \approx 9.49 \) units.
• Pythagorean Theorem: In right-angled triangles, if one leg is \( \sqrt{90} \), you can determine the hypotenuse or the other leg if the lengths of the sides follow the Pythagorean relationship \( a^2 + b^2 = c^2 \).
• Algebraic Solutions: Solving quadratic equations sometimes involves square roots. For example, solving \( x^2 = 90 \) involves finding \( x = \pm \sqrt{90} \).

These examples illustrate how the square root of 90 is applied in various contexts, enhancing both practical problem-solving and theoretical understanding in mathematics and science.

• Is the square root of 90 a rational number?

  No, the square root of 90 is not a rational number. Rational numbers can be expressed as the quotient of two integers, but the square root of 90 is an irrational number because it cannot be exactly expressed as a fraction. The decimal representation of √90 is non-terminating and non-repeating, approximately equal to 9.48683298...

• What is the simplified form of the square root of 90?

  The square root of 90 can be simplified by factoring it into a product of a square number and another integer. Here's how it works:

  • Factor 90 into its prime factors: 90 = 2 × 3² × 5.
  • Identify the perfect square factor: 3².
  • Simplify using the square root of the perfect square: √90 = √(3² × 10) = 3√10.

  Thus, the simplified form of √90 is 3√10.

• How to calculate the square root using a calculator?

  To calculate the square root of 90 using a calculator, follow these steps:

  1. Turn on your calculator and ensure it is in standard mode.
  2. Press the square root (√) button, often represented by √ or an arrow pointing to the top-left.
  3. Enter the number 90.
  4. Press the equal sign (=) or enter button.
  5. The display will show the approximate value of √90, which is about 9.48683298.

• Can the square root of 90 be simplified further?

  The square root of 90 is already simplified to 3√10. This is the simplest form because the remaining factor under the square root, 10, does not contain any perfect square factors that can be simplified further.

The square root of 90, denoted as \( \sqrt{90} \), is an intriguing number with several applications and properties. Calculating it yields approximately \( 9.48683298 \), a non-terminating, non-repeating decimal, which classifies it as an irrational number. In radical form, it simplifies to \( 3\sqrt{10} \).

This number doesn't fit neatly into the category of perfect squares, which means that while its exact value can't be expressed as a simple fraction, it plays a vital role in various mathematical contexts. For example, understanding the square root of 90 is essential in solving problems involving geometry, physics, and engineering where approximations are often needed.

To summarize the key points covered:

• The square root of 90 lies between the perfect squares of 81 and 100, making it a bit more than 9 but less than 10.
• It can be expressed in several forms: as \( 9.48683298 \) (decimal), \( 3\sqrt{10} \) (simplified radical), or \( 90^{1/2} \) (exponential form).
• Despite being an irrational number, the square root of 90 has practical uses in calculations and approximations, making it an important concept in both theoretical and applied mathematics.

Ultimately, whether you're using manual methods, a calculator, or a software tool like Excel, finding the square root of 90 demonstrates the power and beauty of mathematics in handling numbers that aren't perfectly neat and tidy.

By exploring different methods of calculation and simplification, you can appreciate the depth and utility of square roots in mathematical problem-solving.