90 Square Root Simplified: Easy Steps to Simplify √90

The square root of 90 simplified can be represented as:

\[\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\]

Where \( \sqrt{9} = 3 \).

Additional Information:

• Factorization method: \( 90 = 9 \times 10 \)
• Prime factors: \( 90 = 2 \times 3^2 \times 5 \)
• Exact value: \( \sqrt{90} \approx 9.4868 \)

Simplifying square roots is a fundamental skill in mathematics that can help you solve a variety of problems more efficiently. By breaking down a number into its prime factors, we can simplify the square root to its most reduced form. This process not only makes calculations easier but also enhances your understanding of number properties.

Let's start with some basic concepts and a step-by-step guide to simplify the square root of 90.

• Identify the prime factors of the number.
• Pair the prime factors.
• Simplify the expression by taking the square root of the paired factors.

Step-by-Step Guide to Simplifying √90

1. Prime Factorization of 90:
   • 90 can be factored into 9 and 10.
   • 9 can be further factored into 3 and 3.
   • 10 can be factored into 2 and 5.
   • Thus, the prime factors of 90 are 2, 3, 3, and 5.
2. Group the prime factors into pairs:
   • We can pair one 3 with another 3.
   • The remaining factors are 2 and 5, which do not have pairs.
3. Simplify the square root expression:
   • √90 = √(2 × 3 × 3 × 5)
   • √90 = √(3² × 2 × 5)
   • √90 = 3√(2 × 5)
   • √90 = 3√10

By following these steps, we can see that the simplified form of √90 is 3√10. Understanding this method can help you simplify other square roots effectively.

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Square roots are fundamental in various areas of mathematics, including geometry and algebra.

Square roots can be classified into two categories:

• Perfect Squares: Numbers whose square roots are integers. For instance, 1, 4, 9, 16, 25, etc.
• Non-Perfect Squares: Numbers whose square roots are not integers and often involve irrational numbers. For example, 2, 3, 5, 6, 7, etc.

Let's explore how to find the square root of non-perfect squares using the example of 90:

1. Start with the prime factorization of 90:
   • 90 = 2 × 3 × 3 × 5
2. Pair the prime factors:
   • Pair the two 3's: (3 × 3) = 9
   • Leave the remaining 2 and 5 unpaired.
3. Simplify the square root expression using the pairs:
   • \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \)
   • \( \sqrt{90} = \sqrt{3^2 \times 2 \times 5} \)
   • \( \sqrt{90} = 3 \sqrt{2 \times 5} \)
   • \( \sqrt{90} = 3 \sqrt{10} \)

By breaking down the number into its prime factors and pairing them, we simplify the square root of 90 to \( 3 \sqrt{10} \). Understanding this process is essential for simplifying square roots of other non-perfect square numbers as well.

The prime factorization method is a systematic way to break down a number into its prime factors, which are the building blocks of the number. This method is particularly useful for simplifying square roots. Let's use the prime factorization method to simplify the square root of 90 step by step.

1. Identify the number to be factored:
   • We start with 90.
2. Find the smallest prime number that divides 90:
   • 90 is an even number, so it is divisible by 2.
   • \( 90 \div 2 = 45 \)
3. Repeat the process with the quotient (45):
   • 45 is not even, so we try the next smallest prime number, which is 3.
   • \( 45 \div 3 = 15 \)
4. Continue factoring the quotient (15):
   • 15 is divisible by 3.
   • \( 15 \div 3 = 5 \)
   • 5 is a prime number.
5. List all the prime factors of 90:
   • So, the prime factorization of 90 is \( 2 \times 3 \times 3 \times 5 \).

Now that we have the prime factors, we can use them to simplify the square root:

\( \sqrt{90} = \sqrt{2 \times 3 \times 3 \times 5} \)

Pair the prime factors to simplify:

\( \sqrt{90} = \sqrt{3^2 \times 2 \times 5} \)

Extract the square of the paired factors:

\( \sqrt{90} = 3 \sqrt{2 \times 5} \)

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

Using the prime factorization method, we have simplified the square root of 90 to \( 3 \sqrt{10} \). This method is efficient and helps in understanding the composition of numbers.

To simplify the square root of 90, we can use the prime factorization method. Here are the steps:

1. Start by finding the prime factorization of 90. We break down 90 into its prime factors:

   \[ 90 = 2 \times 45 \]

   \[ 45 = 3 \times 15 \]

   \[ 15 = 3 \times 5 \]

   So, the prime factorization of 90 is:

   \[ 90 = 2 \times 3^2 \times 5 \]

2. Next, we write the square root of 90 using its prime factorization:

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

3. We can separate the square root of the product into the product of the square roots:

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

4. Simplify the square roots of the perfect squares:

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

   So, we have:

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

5. Combine the simplified parts:

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

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

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

To simplify the square root of 90, we start by breaking down 90 into its prime factors. Here is a step-by-step guide:

1. First, identify the prime factors of 90.
2. 90 can be divided by 2, the smallest prime number:
   • 90 ÷ 2 = 45
3. Next, 45 can be divided by 3, the next smallest prime number:
   • 45 ÷ 3 = 15
4. Then, 15 can be divided by 3 again:
   • 15 ÷ 3 = 5
5. Finally, 5 is a prime number.

So, the prime factorization of 90 is:

\[ 90 = 2 \times 3 \times 3 \times 5 \]

We can group the prime factors into pairs to simplify the square root:

\[ 90 = 2 \times 3^2 \times 5 \]

Taking the square root of both sides, we get:

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

Since \(\sqrt{3^2} = 3\), we can simplify this to:

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

Therefore, the simplified form is:

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

To simplify the square root of 90, we need to identify and pair the prime factors of 90. Here is a step-by-step guide:

1. First, find the prime factors of 90:
   • 90 can be divided by 2 (the smallest prime number): \(90 \div 2 = 45\)
   • Next, divide 45 by 3 (the next smallest prime number): \(45 \div 3 = 15\)
   • Finally, divide 15 by 3 again: \(15 \div 3 = 5\)
2. So, the prime factorization of 90 is:

   \[ 90 = 2 \times 3 \times 3 \times 5 \]

3. Identify the pairs of prime factors:

   From the factorization, we see there is a pair of 3s. This pair can be taken out of the square root.

4. Simplify the square root:
   • We can take the pair of 3s out as a single 3: \(\sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3\sqrt{2 \times 5}\)
   • Multiply the remaining factors under the square root: \(\sqrt{2 \times 5} = \sqrt{10}\)
5. Thus, the simplified form of \(\sqrt{90}\) is:

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

By following these steps, we can simplify the square root of 90 efficiently and accurately.

To simplify the square root of 90, we follow these steps:

1. First, perform prime factorization of 90 to find its prime factors:
• 90 = 2 × 3² × 5
2. Next, express the square root of 90 in terms of its prime factors:
• √90 = √(2 × 3² × 5)
3. Separate the factors under the square root:
• √90 = √2 × √3² × √5
4. Take the square root of any perfect squares:
• √3² = 3
5. Multiply the simplified factors:
• √90 = 3 × √(2 × 5)
• √90 = 3√10

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

In decimal form, √90 is approximately 9.486.

Visualizing the simplification of the square root of 90 can help in understanding the process more clearly. Here, we will use a factor tree and a step-by-step diagram to illustrate the simplification.

Factor Tree

Let's start by breaking down 90 into its prime factors:

• 90 can be divided by 2 (since it is even):
• 90 ÷ 2 = 45
• 45 can be divided by 3 (since the sum of its digits is 9, which is divisible by 3):
• 45 ÷ 3 = 15
• 15 can be divided by 3 again:
• 15 ÷ 3 = 5

So, the prime factorization of 90 is:

$$

90
=
2
×
3
×
3
×
5

Simplification Diagram

Next, we simplify the square root of 90 by grouping the prime factors:

• Group the pairs of the same prime numbers:
• $\sqrt{2\times 3\times 3\times 5}$
• Identify pairs:
• The pair of 3s can be taken out of the square root:
• $\sqrt{3²\times 2\times 5}=3\sqrt{2\times 5}$

Final Simplified Form

Thus, the simplified form of the square root of 90 is:

$$

3

10

This can also be approximately calculated as:

$$

3
×
3.162
≈
9.486

Diagram

When simplifying the square root of 90, it's important to avoid common mistakes that can lead to incorrect results. Here are some key points to keep in mind:

• Incorrect Prime Factorization: Ensure that you correctly identify the prime factors of 90. The correct factorization is \( 90 = 2 \times 3^2 \times 5 \). Mistakes in factorization can lead to incorrect simplification.
• Missing Pairs of Prime Factors: When simplifying square roots, only pairs of identical prime factors can be taken out from under the radical. For \( \sqrt{90} \), you can take out \( 3 \) (since \( 3^2 \) is a pair), but not the 2 or 5.
• Incorrect Simplification: Always remember to simplify the square root fully. For \( \sqrt{90} \), the correct simplification is \( 3\sqrt{10} \), not \( \sqrt{9 \times 10} \) or other incorrect forms.
• Ignoring Simplification Steps: Ensure that you follow all steps in the simplification process. Skipping steps can lead to errors. For instance:
  1. Factorize \( 90 \) into its prime factors: \( 90 = 2 \times 3^2 \times 5 \).
  2. Group the pairs: \( 3^2 \) can be taken out of the square root.
  3. Simplify: \( \sqrt{90} = \sqrt{3^2 \times 10} = 3\sqrt{10} \).
• Incorrect Use of Calculators: If using a calculator, make sure to use the square root function correctly. Some calculators require you to input the number first and then press the square root button, while others do the reverse.
• Misinterpreting Results: After simplification, ensure that you interpret the result correctly. For example, \( 3\sqrt{10} \approx 9.4868 \), but \( 3\sqrt{10} \) is the exact simplified form.

By keeping these common mistakes in mind, you can simplify the square root of 90 accurately and confidently.

The process of simplifying square roots has many practical applications across various fields. Here are some examples:

• Finance:

  In finance, square roots are used to calculate stock market volatility, a measure of how much a stock's price fluctuates over time. This helps investors assess the risk of their investments.

• Architecture and Engineering:

  Square roots are used to determine the natural frequency of structures like bridges and buildings. This helps engineers predict how structures will respond to different loads, such as wind or traffic.

• Science:

  In scientific calculations, square roots are used to determine various quantities, such as the velocity of an object, the amount of radiation absorbed by a material, or the intensity of sound waves.

• Statistics:

  In statistics, the standard deviation, which measures how spread out data is from the mean, is calculated using square roots. This helps statisticians analyze data and make informed decisions.

• Geometry:

  Geometry frequently involves square roots, such as when calculating the length of the sides of a right triangle using the Pythagorean theorem or finding the side length of a square given its area.

• Computer Science:

  Square roots are used in various computer algorithms, including those for encryption, image processing, and game physics. These applications are critical for secure data transmission and realistic graphics.

• Navigation:

  Square roots are essential in navigation for computing distances between points on a map. Pilots, for example, use these calculations to determine flight paths and distances.

• Electrical Engineering:

  In electrical engineering, square roots are used to compute power, voltage, and current in circuits. These calculations are crucial for designing and developing efficient electrical systems.

These examples demonstrate the importance and versatility of simplifying square roots in solving real-world problems.

Here are a few additional examples and practice problems to help reinforce your understanding of simplifying square roots:

• Example 1: Simplify \( \sqrt{72} \)

  Step-by-step solution:

  1. Prime factorize 72: \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \)
  2. Group the prime factors into pairs: \( \sqrt{72} = \sqrt{(2 \times 2) \times (3 \times 3) \times 2} \)
  3. Take the pairs out of the square root: \( \sqrt{72} = 2 \times 3 \times \sqrt{2} \)
  4. Simplified form: \( \sqrt{72} = 6\sqrt{2} \)

• Example 2: Simplify \( \sqrt{108} \)

  Step-by-step solution:

  1. Prime factorize 108: \( 108 = 2 \times 2 \times 3 \times 3 \times 3 \)
  2. Group the prime factors into pairs: \( \sqrt{108} = \sqrt{(2 \times 2) \times (3 \times 3) \times 3} \)
  3. Take the pairs out of the square root: \( \sqrt{108} = 2 \times 3 \times \sqrt{3} \)
  4. Simplified form: \( \sqrt{108} = 6\sqrt{3} \)

• Practice Problems:

  • 1. Simplify \( \sqrt{50} \)
  • 2. Simplify \( \sqrt{200} \)
  • 3. Simplify \( \sqrt{18} \)
  • 4. Simplify \( \sqrt{45} \)
  • 5. Simplify \( \sqrt{98} \)

  Solutions:

  • 1. \( \sqrt{50} = 5\sqrt{2} \)
  • 2. \( \sqrt{200} = 10\sqrt{2} \)
  • 3. \( \sqrt{18} = 3\sqrt{2} \)
  • 4. \( \sqrt{45} = 3\sqrt{5} \)
  • 5. \( \sqrt{98} = 7\sqrt{2} \)

Simplifying the square root of 90 involves several straightforward steps. By understanding the prime factorization method, we were able to break down 90 into its prime factors, identify pairs, and simplify the expression. The simplified form, \( \sqrt{90} = 3\sqrt{10} \), not only helps in making calculations easier but also provides a clearer understanding of the square root's composition.

Knowing how to simplify square roots is a fundamental skill in algebra that has practical applications in various mathematical problems, including geometry and calculus. The process we followed ensures accuracy and reinforces the importance of prime factorization and proper simplification techniques. By avoiding common mistakes and practicing with additional examples, you can become proficient in simplifying square roots and apply these skills to solve more complex mathematical challenges.

Ultimately, mastering this concept opens the door to deeper mathematical understanding and problem-solving abilities, making it a valuable tool in your mathematical toolkit.