Simplify Square Root of 90: Easy Steps to Master the Concept

The square root of 90 can be simplified by breaking it down into its prime factors. Here is a detailed step-by-step process to simplify √90.

Step-by-Step Process

1. Identify the prime factors of 90:

   • 90 = 2 × 3² × 5

2. Rewrite 90 under the square root as the product of its prime factors:

   \[\sqrt{90} = \sqrt{2 \times 3^2 \times 5}\]

3. Separate the perfect square (3²) from the other factors:

   \[\sqrt{2 \times 3^2 \times 5} = \sqrt{3^2} \times \sqrt{2 \times 5} = 3 \sqrt{10}\]

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

Alternative Methods

There are various methods to approximate or simplify the square root of 90:

• Using the long division method, the square root of 90 can be approximated to three decimal places: \(\sqrt{90} \approx 9.486\).
• Recognizing that 90 lies between the perfect squares of 81 and 100, thus \(\sqrt{90}\) is between 9 and 10.

Applications

Here are some practical applications of simplifying the square root of 90:

1. Example 1: Fencing a Square Garden

   If the area of a square garden is 90 square yards, the length of the side is \(\sqrt{90}\) yards, approximately 9.486 yards. The total length of the fence required would be \(4 \times 9.486 = 37.944\) yards.

2. Example 2: Simplifying Expressions

   To simplify \(\sqrt{90} \times \sqrt{75}\):

   \(\sqrt{90} \times \sqrt{75} = 3 \sqrt{10} \times 5 \sqrt{10} = 15 \times 10 = 150\)

By understanding and applying these methods, you can simplify and work with the square root of 90 more effectively.

Simplifying square roots is an essential skill in mathematics, providing a more manageable form of radical expressions. By breaking down a number into its prime factors and identifying perfect squares, you can simplify the square root to its simplest form. This process is useful for solving equations, simplifying expressions, and understanding the properties of numbers.

Here’s a step-by-step guide to help you simplify square roots:

1. Identify the prime factors: Break down the number under the square root into its prime factors. For example, 90 can be factored into 2 × 3² × 5.

2. Group the perfect squares: Identify and group the perfect square factors. In the case of 90, 3² is a perfect square.

3. Rewrite the expression: Separate the perfect square from the other factors under the square root:

   \[\sqrt{90} = \sqrt{2 \times 3^2 \times 5}\]

4. Simplify the perfect squares: Take the square root of the perfect square factor:

   \[\sqrt{3^2} = 3\]

5. Combine the results: Combine the simplified perfect square with the remaining factors under the square root:

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

By following these steps, you can simplify the square root of any number, making it easier to work with in various mathematical contexts.

Simplifying the square root of 90 involves breaking down the number into its prime factors and simplifying from there. Follow these steps to simplify √90:

1. Identify the prime factors of 90.
• 90 can be factored into 9 and 10, i.e., \( 90 = 9 \times 10 \).
• Next, factor 9 and 10 into their prime factors: \( 9 = 3^2 \) and \( 10 = 2 \times 5 \).
2. Rewrite the square root using these factors:
• \( \sqrt{90} = \sqrt{3^2 \times 10} \).
3. Simplify by taking the square root of the perfect squares:
• \( \sqrt{3^2 \times 10} = \sqrt{3^2} \times \sqrt{10} = 3\sqrt{10} \).

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

Additionally, the decimal approximation of \( \sqrt{90} \) is approximately 9.49.

The prime factorization method is a straightforward approach to simplifying square roots. This method involves breaking down a number into its prime factors and then pairing these factors to simplify the square root. Below are the detailed steps to simplify the square root of 90 using prime factorization:

1. Factorize the Number into Prime Factors:

   Start by dividing 90 by the smallest prime number, which is 2, and continue the process with the quotient until you cannot divide further. The prime factorization of 90 is:

   • 90 ÷ 2 = 45
   • 45 ÷ 3 = 15
   • 15 ÷ 3 = 5
   • 5 ÷ 5 = 1

   So, the prime factors of 90 are \(2 \times 3^2 \times 5\).

2. Organize the Prime Factors into Pairs:

   Group the identical factors into pairs:

   • Pair of 3s: \(3 \times 3\)
   • Unpaired factors: 2 and 5
3. Multiply One Element from Each Pair:

   For each pair of identical primes, select one prime from the pair and multiply them together:

   \(3 \times \sqrt{2 \times 5}\)

4. Calculate the Simplified Square Root:

   Combine the results to get the simplified square root:

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

This method simplifies the square root by systematically breaking down the number into manageable parts and leveraging the properties of prime factors.

The Long Division Method is a reliable technique to find the square root of any number, whether it is a perfect square or not. Here, we will simplify the square root of 90 using this method. Follow these steps for a detailed understanding:

1. Start by writing the number 90 as 90.000000 (adding decimal places to allow for precision).
2. Group the digits in pairs from the decimal point, moving both left and right (90.000000).
3. Find the largest number whose square is less than or equal to the first group (90). Here, 9² = 81, which is the closest. Write 9 as the first digit of the quotient.
4. Subtract 81 from 90, leaving a remainder of 9. Bring down the next pair of zeros, making the remainder 900.
5. Double the quotient (9) to get 18, and write it as the new divisor's first digit. Determine a digit (x) such that 18x * x is less than or equal to 900. The number 4 fits, as 184 * 4 = 736.
6. Write 4 in the quotient next to 9 (making it 9.4) and subtract 736 from 900, giving a remainder of 164. Bring down the next pair of zeros, making it 16400.
7. Double the current quotient (94), giving 188. Determine a digit (y) such that 188y * y is less than or equal to 16400. The digit 8 works since 1888 * 8 = 15104.
8. Continue this process until you reach the desired level of accuracy. For example, after a few more steps, you might find that √90 ≈ 9.4868.

Using this method ensures you get a precise and accurate square root. The long division method is particularly useful when dealing with non-perfect squares and provides a systematic approach to obtaining the square root step-by-step.

Approximating the square root of 90 can be done through a few methods. One of the simplest ways is by making an educated guess and refining it through iteration. Here’s a detailed step-by-step process:

1. Choose an initial estimate. Since \( \sqrt{81} = 9 \) and \( \sqrt{100} = 10 \), an initial guess for \( \sqrt{90} \) could be 9.5.
2. Refine the estimate using the average method: Take the initial guess (9.5) and divide 90 by it. Then average the result with the initial guess.
   • \(\text{New estimate} = \frac{9.5 + \frac{90}{9.5}}{2} \)
3. Repeat the process: Use the new estimate to further refine the calculation.
   • For example, if the new estimate is 9.487, calculate \(\frac{9.487 + \frac{90}{9.487}}{2} \).
4. Stop when the difference between the estimates is sufficiently small. This iterative process can quickly converge to the actual square root of 90.

Another method is using the long division method to approximate the square root. This traditional technique involves dividing the number into pairs of digits from the decimal point and finding each digit of the root step-by-step.

By following these methods, you can approximate the square root of 90 to a high degree of accuracy, often to several decimal places.

The square root of 90, approximately 9.4868, has numerous practical applications across various fields. Here are some ways it is used in real-world scenarios:

• Finance: In finance, the square root is used to calculate stock market volatility. This involves taking the square root of the variance of stock returns, helping investors assess the risk of their investments.
• Architecture: Engineers use square roots to determine the natural frequency of structures like bridges and buildings. This helps predict how structures will respond to different loads, ensuring their stability and safety.
• Science: Scientific calculations often involve square roots, such as determining the velocity of moving objects, the amount of radiation absorbed by materials, and the intensity of sound waves. These calculations aid in understanding and developing new technologies.
• Statistics: In statistics, square roots are used to compute standard deviation, which is the square root of the variance. This metric helps in analyzing data sets and making informed decisions based on statistical analysis.
• Geometry: Square roots are essential in geometry for calculating the area and perimeter of shapes, solving problems involving right triangles, and applying the Pythagorean theorem.
• Computer Science: In computer programming, square roots are used in encryption algorithms, image processing, and game physics, playing a crucial role in securing data and creating realistic simulations.
• Navigation: Navigators use square roots to compute distances between points on maps or globes, and to estimate courses and directions, which is vital for accurate travel and exploration.
• Electrical Engineering: Electrical engineers use square roots to compute power, voltage, and current in circuits. These calculations are critical for designing and developing electrical systems and devices.

Simplifying the square root of 90 can be done using various methods. Here, we will explore some examples step by step.

Example 1: Simplifying √90 Using Prime Factorization

To simplify √90 using prime factorization, follow these steps:

1. Break 90 into its prime factors:
• 90 = 2 × 3 × 3 × 5
2. Group the prime factors into pairs of the same number:
• 90 = 3² × 2 × 5
3. Take the square root of each pair of factors and simplify:
• √(3² × 2 × 5) = 3√(2 × 5) = 3√10

Therefore, √90 simplifies to 3√10.

Example 2: Using the Product Property of Square Roots

The product property of square roots states that √(a × b) = √a × √b. We can use this property to simplify √90:

1. Rewrite 90 as a product of two factors, one of which is a perfect square:
• 90 = 9 × 10
2. Apply the product property:
• √90 = √(9 × 10) = √9 × √10
3. Simplify the square root of the perfect square:
• √9 = 3
4. Combine the simplified square root with the other factor:
• √90 = 3√10

Thus, using the product property, √90 simplifies to 3√10.

Example 3: Verification by Calculation

Let's verify the simplification by calculating both the simplified and original forms:

• Original form: √90 ≈ 9.486
• Simplified form: 3√10 ≈ 3 × 3.162 = 9.486

Both forms yield the same result, confirming that √90 simplifies correctly to 3√10.

Example 4: Simplifying √90 in Radical Form

To express the square root of 90 in its simplest radical form:

1. Factor 90 into prime factors:
• 90 = 2 × 3² × 5
2. Group the prime factors:
• 90 = 3² × 10
3. Take out the square root of the grouped factor:
• √(3² × 10) = 3√10

Therefore, the simplest radical form of √90 is 3√10.

These examples demonstrate various methods to simplify the square root of 90, all leading to the same simplified form: 3√10.

Simplifying square roots can be tricky, and several common mistakes can lead to incorrect results. Here are some pitfalls to watch out for and tips to help you avoid them:

• Forgetting to Check for Perfect Squares:

  Always check if the number inside the square root is a perfect square. This can greatly simplify the process. For example, \(\sqrt{90}\) can be broken down into \(\sqrt{9 \times 10}\) which simplifies to \(3\sqrt{10}\).

• Rushing Through the Process:

  Simplifying square roots requires careful step-by-step work. Take your time to factorize the number correctly and simplify each part methodically. Hurrying can lead to missed factors or incorrect simplification.

• Ignoring Prime Factorization:

  Prime factorization is a reliable method for simplifying square roots. Breaking the number down into its prime factors helps identify perfect squares easily. For instance, \(90 = 2 \times 3^2 \times 5\), and thus \(\sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3\sqrt{10}\).

• Incorrect Application of Simplification Rules:

  Ensure you apply the rule \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\) correctly. Misapplying this rule can result in errors. For example, \(\sqrt{90} \neq \sqrt{9} \times \sqrt{10} = 3 \times 10\). It should be \(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\).

• Leaving Terms Inside the Square Root:

  Whenever possible, simplify the terms inside the square root. If left unchecked, this can make the expression more complex and harder to interpret.

By keeping these common mistakes in mind, you can simplify square roots more accurately and efficiently.

Simplifying the square root of 90 can be approached using various advanced methods that provide deeper insights into the mathematical process. Here are some advanced techniques:

Prime Factorization Method

This method involves breaking down the number into its prime factors and simplifying accordingly:

1. Start by expressing 90 as a product of its prime factors: \( 90 = 2 \times 3^2 \times 5 \).
2. Group the prime factors in pairs where possible. Here, \( 3^2 \) forms a pair.
3. Take one number from each pair out of the square root: \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3\sqrt{2 \times 5} \).
4. Combine the remaining factors inside the square root: \( 3\sqrt{10} \).

Thus, the simplified form is \( 3\sqrt{10} \).

Using Exponential Form

This method utilizes the properties of exponents to simplify the square root:

1. Express the number under the square root in exponential form: \( 90 = 9 \times 10 = 3^2 \times 10 \).
2. Apply the property of square roots to separate the factors: \( \sqrt{90} = \sqrt{3^2 \times 10} \).
3. Simplify by taking the square root of the perfect square: \( \sqrt{3^2} = 3 \), so \( \sqrt{90} = 3\sqrt{10} \).

Long Division Method

The long division method provides a step-by-step manual calculation:

1. Set up 90 under the long division symbol and pair the digits starting from the decimal point (90.00 00).
2. Find a number whose square is closest to 90 without exceeding it (9 × 9 = 81). Subtract 81 from 90, bringing down the next pair of zeros to get 900.
3. Double the initial quotient (9) to get 18, and use this as the new divisor (180_). Find a digit to complete the divisor (184), ensuring 184 × 4 = 736 is as close to 900 as possible. Subtract to get 164.
4. Bring down the next pair of zeros (16400) and repeat the process until the desired precision is achieved.

After a few iterations, the result approximates \( \sqrt{90} \approx 9.486 \).

Approximations Using Calculators

For quick and precise results, using a scientific calculator is effective:

• Enter 90 and press the square root button.
• The calculator provides an immediate result, \( \sqrt{90} \approx 9.486 \).

Application in Real Numbers

Understanding the irrational nature of \( \sqrt{90} \) can further simplify calculations:

• \( \sqrt{90} \) cannot be expressed as a simple fraction, making it an irrational number.
• The decimal expansion is non-repeating and non-terminating.

These advanced methods enhance the precision and understanding of simplifying square roots, providing valuable tools for more complex mathematical applications.

Interactive tools and visual aids can significantly enhance the understanding and simplification of square roots, including √90. Here are some useful resources and tools:

Interactive Calculators

• Square Root Calculator: Online calculators such as those available on allow you to input any number and see the step-by-step simplification process. For √90, it shows the factorization and simplification as √(9×10) = 3√10.

• Symbolab Calculator: This tool not only simplifies square roots but also provides visual explanations and graphs. You can enter √90, and it breaks down the steps to 3√10, showing the prime factorization and the separation of perfect squares.

Visual Aids

• Prime Factorization Trees: Drawing a prime factorization tree helps in visually understanding the simplification process. For 90, the tree shows how it breaks down into 2 × 3² × 5, highlighting the perfect square 3².

• Graphical Representations: Websites like offer graphical representations of square roots and their simplifications. These visuals aid in comprehending the concept of square roots by displaying geometric interpretations.

Step-by-Step Tutorials

• Video Tutorials: Platforms like YouTube have numerous tutorials that visually explain the process of simplifying square roots. Watching these videos can provide a clearer understanding of the steps involved in simplifying √90.

• Interactive Lessons: Websites such as offer interactive lessons where you can practice simplifying square roots with immediate feedback, reinforcing the learning process.

Practice Tools

• Worksheets and Quizzes: Printable worksheets and online quizzes from educational websites help practice the simplification of square roots. These resources often include step-by-step solutions to problems like simplifying √90.

• Interactive Games: Math games that incorporate square root simplification can make learning fun and engaging. These games often include challenges that require the player to simplify square roots correctly to advance.

Using these interactive tools and visual aids can greatly enhance your understanding and ability to simplify square roots, making the learning process both effective and enjoyable.

• What is the square root of 90 simplified?

  The square root of 90 simplified in radical form is \( 3\sqrt{10} \).

• Is the square root of 90 a rational number?

  No, the square root of 90 is an irrational number. When expressed in decimal form, it is approximately 9.486832980505138, which does not terminate or repeat.

• How can we calculate the square root of 90?

  You can calculate the square root of 90 using various methods such as the prime factorization method or the long division method. The prime factorization method involves breaking down 90 into its prime factors and simplifying. The long division method involves iteratively finding the digits of the square root.

• Between which two numbers does the square root of 90 lie?

  The square root of 90 lies between 9 and 10, as it is more than 9² (81) and less than 10² (100).

• What is the square root of 90 rounded to the nearest tenth?

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

• What is the square root of 90 in exponential form?

  The square root of 90 in exponential form is \( 90^{1/2} \).

• Can you provide an example of simplifying the square root of 90?

  Sure! Here is a step-by-step simplification:

  1. Factor 90 into its prime factors: \( 90 = 2 \times 3^2 \times 5 \).
  2. Group the prime factors: \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \).
  3. Take out the square root of the perfect squares: \( \sqrt{90} = 3\sqrt{10} \).

  Thus, \( \sqrt{90} = 3\sqrt{10} \).

For further learning and to deepen your understanding of simplifying the square root of 90, the following resources and references are highly recommended:

• Mathway:

  Mathway offers a comprehensive step-by-step breakdown of the simplification process. This tool allows you to enter the problem and see detailed solutions with explanations.

• Symbolab:

  Symbolab provides an in-depth guide with examples and solutions for simplifying square roots. It includes various mathematical tools and calculators to help you practice and understand the concept better.

• Khan Academy:

  Khan Academy offers free video tutorials and practice exercises on simplifying square roots. The platform explains the concepts clearly and provides numerous examples to work through.

• MathIsFun:

  This website breaks down mathematical concepts into easy-to-understand lessons. It is particularly useful for beginners who need a basic introduction to simplifying square roots.

These resources will provide you with a variety of approaches and tools to master the simplification of square roots, including √90. Explore these links to enhance your learning experience and tackle more complex mathematical problems.