Square Root of 90 Simplified: Easy Steps to Simplify √90

The square root of 90 can be simplified by finding the prime factorization of 90 and then simplifying the expression. The prime factorization of 90 is:

90 = 2 × 3² × 5

To simplify the square root of 90, we can rewrite it using the prime factors:

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

Since the square root of a product is the product of the square roots, and \(\sqrt{3^2} = 3\), we can simplify further:

\(\sqrt{90} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5} = 3 \sqrt{2 \times 5} = 3 \sqrt{10}\)

Steps to Simplify

1. Find the prime factorization of the number.
2. Group the prime factors into pairs.
3. Bring out the square roots of the pairs from under the radical.
4. Multiply the remaining factors under the radical and simplify.

Visual Representation

The visual representation of simplifying the square root of 90 is shown below:

Conclusion

The simplified form of the square root of 90 is \(3 \sqrt{10}\). This method can be used to simplify the square roots of other numbers by following similar steps of prime factorization and simplification.

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 √. For example, the square root of 9 is 3, because 3 × 3 = 9.

Square roots are essential in various fields of mathematics and science, including algebra, geometry, and physics. Understanding square roots helps in solving quadratic equations, analyzing geometric shapes, and even in everyday calculations such as determining areas and volumes.

There are a few key points to understand about square roots:

• Every positive number has two square roots: one positive and one negative. For instance, the square roots of 16 are 4 and -4.
• The square root of zero is zero.
• Negative numbers do not have real square roots because no real number squared gives a negative result. However, they have imaginary square roots, which are discussed in complex number theory.

Square roots can be represented in various forms, including exact form, decimal form, and simplified radical form. Simplifying square roots involves expressing the radical in its simplest form, often using prime factorization.

In the following sections, we will explore the concept of square roots in detail, starting with the process of prime factorization and moving on to the steps involved in simplifying the square root of 90.

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Understanding prime factorization is crucial for simplifying square roots.

To perform prime factorization, follow these steps:

1. Start with the smallest prime number: Begin dividing the number by the smallest prime number, which is 2. If the number is even, it is divisible by 2. If not, try the next smallest prime number, which is 3, and so on.
2. Divide the number: Continue dividing the number by the prime number until it is no longer divisible. Each time you divide, write down the prime number used.
3. Move to the next prime number: Once the number is no longer divisible by the current prime number, move to the next smallest prime number and repeat the process.
4. Continue until the number is 1: Keep dividing until the original number has been completely broken down into prime factors, and the final quotient is 1.

Let’s use prime factorization to break down 90:

• 90 is even: Divide by 2: \( 90 \div 2 = 45 \)
• 45 is divisible by 3: Divide by 3: \( 45 \div 3 = 15 \)
• 15 is divisible by 3: Divide by 3: \( 15 \div 3 = 5 \)
• 5 is a prime number: Divide by 5: \( 5 \div 5 = 1 \)

The prime factors of 90 are 2, 3, 3, and 5. We can express 90 as:

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

Prime factorization helps in simplifying the square root by grouping the prime factors into pairs. In the next section, we will use this prime factorization to simplify the square root of 90.

To simplify the square root of 90, we will use the prime factorization method. The goal is to express the square root in its simplest radical form by breaking down the number into its prime factors and pairing them. Here are the steps:

1. Prime Factorization: Begin with the prime factorization of 90. As we found earlier:

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

2. Pair the Factors: Look for pairs of the same number among the prime factors. In this case, the number 3 appears twice, forming a pair:

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

3. Simplify Inside the Square Root: Each pair of factors can be taken out of the square root as a single factor. The pair of 3s can be simplified as follows:

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

4. Take the Pair Out: Take the square root of the pair of 3s (which is 3) out of the square root:

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

5. Simplify the Remaining Product: Multiply the remaining numbers inside the square root:

   \[
   \sqrt{10}
   \]

6. Combine the Results: Combine the simplified factor outside the square root with the remaining simplified radical:

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

Therefore, the simplified form of the square root of 90 is \( 3 \sqrt{10} \). This process helps in understanding how to break down and simplify square roots using prime factorization and pairing factors.

Simplifying square roots involves expressing the radical in its simplest form. The process typically includes prime factorization, pairing factors, and simplifying the expression. Here are the detailed steps to simplify square roots:

1. Find the Prime Factorization: Begin by finding the prime factors of the number under the square root. Prime factorization breaks down a number into a product of prime numbers.
   • Example: To simplify \( \sqrt{90} \), first find its prime factors: \( 90 = 2 \times 3 \times 3 \times 5 \).
2. Group the Prime Factors: Group the prime factors into pairs. Each pair of identical factors can be taken out of the square root as a single factor.
   • Example: In \( \sqrt{90} = \sqrt{2 \times 3 \times 3 \times 5} \), the factors can be grouped as \( \sqrt{2 \times (3 \times 3) \times 5} \).
3. Simplify Inside the Square Root: Simplify the expression inside the square root by identifying and taking out pairs.
   • Example: Since \( 3 \times 3 = 3^2 \), we can take out 3 as \( \sqrt{2 \times 3^2 \times 5} = 3 \sqrt{2 \times 5} \).
4. Multiply the Remaining Factors: Multiply the remaining numbers inside the square root to form a new simplified radical.
   • Example: Inside the square root, multiply \( 2 \times 5 = 10 \). This gives \( 3 \sqrt{10} \).
5. Combine the Results: Write the final simplified form by combining the factor outside the square root with the simplified radical.
   • Example: The simplified form of \( \sqrt{90} \) is \( 3 \sqrt{10} \).

Following these steps ensures that you can simplify any square root efficiently. Practice with different numbers to become more familiar with the process and improve your skills in simplifying radicals.

Let's simplify the square root of 90 step by step:

1. Start with the prime factorization of 90: \( 90 = 2 \times 3^2 \times 5 \).
2. Separate the factors into pairs of the same number inside the square root: \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \).
3. Take out the square of the prime factors from under the square root sign: \( \sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3\sqrt{10} \).

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

Let's visualize the simplification of \( \sqrt{90} \) using its prime factorization:

By following these steps, we simplify \( \sqrt{90} \) to \( 3\sqrt{10} \).

When simplifying the square root of 90, be mindful of the following common mistakes:

• Incorrect prime factorization: Ensure 90 is correctly factored into its prime components (2, 3, and 5).
• Missing or incorrect grouping of factors: Group the prime factors properly under the square root.
• Forgetting to simplify the square of factors outside the square root.
• Incorrect final simplified form: Double-check the result to ensure it matches the simplified expression \( 3\sqrt{10} \).
• Confusing steps: Follow the step-by-step process to avoid mixing up the order of operations.

By avoiding these mistakes, you can accurately simplify \( \sqrt{90} \) to \( 3\sqrt{10} \).

The simplified form \( 3\sqrt{10} \) for \( \sqrt{90} \) has practical applications in various mathematical contexts:

1. Calculation Efficiency: Simplified square roots allow for quicker calculations in fields like engineering and physics.
2. Problem Solving: It aids in solving equations involving square roots of numbers like 90 in a more manageable form.
3. Education: Understanding the simplification process helps in teaching fundamental concepts of square roots and prime factorization.
4. Real-World Applications: Used in areas such as geometry, where simplified forms are easier to work with in measurements and calculations.

By applying \( 3\sqrt{10} \), we enhance mathematical efficiency and comprehension across various disciplines.

In conclusion, simplifying the square root of 90, we found that \( \sqrt{90} \) can be expressed as \( 3\sqrt{10} \). This simplified form is achieved through the process of prime factorization and extracting the square of the factors outside the square root. Understanding and applying this method not only aids in computational efficiency but also enhances conceptual clarity in mathematical reasoning and problem-solving. By avoiding common mistakes and recognizing the applications of simplified square roots, we can leverage this knowledge across various disciplines, from education to practical applications in engineering and sciences.