Square Root of 90 in Simplest Radical Form: A Complete Guide

The square root of 90 can be simplified into its simplest radical form. Here is a step-by-step explanation of how this is done:

Step-by-Step Process

1. List the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
2. Identify the perfect squares from the list of factors: 1, 9.
3. Divide 90 by the largest perfect square: 90 ÷ 9 = 10.
4. Calculate the square root of the largest perfect square: √9 = 3.
5. Combine the results to get the simplest form: 3√10.

Thus, the square root of 90 in its simplest radical form is $\sqrt{90}=3\sqrt{10}$.

Decimal and Exponential Forms

The square root of 90 can also be expressed in other forms:

• Decimal form: ≈ 9.48683
• Exponential form: 90^1/2 = 3 × 10^1/2

Examples and Applications

Let's look at a couple of examples and applications:

• Example 1: Evaluating √90 × √75.
• Simplified: 3√10 × 5√10 = 15 × 10 = 150
• Example 2: Calculating the perimeter of a square garden with an area of 90 sq yards.
• Side of the garden: √90 ≈ 9.48 yards
• Perimeter: 4 × 9.48 = 37.92 yards

Understanding the Process

Breaking down the steps further helps in understanding the simplification process. For instance:

• We identify pairs of numbers whose product is 90 and simplify.
• Using the average method, we approximate the square root between the perfect squares 81 and 100.

Conclusion

By following these steps, we can simplify the square root of 90 efficiently. This method can be applied to other numbers to find their simplest radical forms as well.

The square root of 90 in simplest radical form is a fundamental concept in algebra. Simplifying this expression involves understanding the factors and applying the properties of square roots. By expressing the square root of 90 in its simplest form, we get a clear and concise representation that is useful in various mathematical contexts.

To simplify the square root of 90, we begin by identifying its factors and finding the largest perfect square factor. This process helps in breaking down the number into more manageable parts and expressing it in the most simplified radical form.

In this article, we will explore the step-by-step process of simplifying the square root of 90, understand its properties, and see practical applications of this mathematical concept.

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, the square root of 90 is written as √90. This value is not a whole number, so it is considered an irrational number.

To simplify the square root of 90, we need to find its simplest radical form. This involves factoring 90 into its prime factors and looking for pairs of factors.

Let's break it down step by step:

• First, factor 90 into prime factors: 90 = 2 × 3² × 5.
• Next, identify pairs of factors that can be taken out of the square root. Here, we have a pair of 3s.
• Simplify by taking the pair out of the radical: √90 = √(3² × 10) = 3√10.

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

Mathematically, this is expressed as:

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

In decimal form, the square root of 90 is approximately 9.4868, but the simplified radical form 3√10 is preferred for exact mathematical representation.

Understanding these basics helps in simplifying and working with square roots in various mathematical problems and applications.

To simplify the square root of 90 in its simplest radical form, follow these steps:

1. Prime Factorization: Start by finding the prime factors of 90. The prime factorization of 90 is:

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

2. Group the Factors: Pair the factors that can form perfect squares. In this case, \( 3^2 \) is a perfect square.

3. Simplify: Rewrite the square root of 90 using the paired factors:

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

   Extract the square root of the perfect square:

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

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

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

To simplify the square root of 90 into its simplest radical form, follow these detailed steps:

1. List the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
2. Identify the perfect squares from these factors: 1, 9.
3. Divide 90 by the largest perfect square found, which is 9: \( \frac{90}{9} = 10 \).
4. Take the square root of the largest perfect square: \( \sqrt{9} = 3 \).
5. Express the original square root in terms of the simplified form: \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10} \).

Therefore, the square root of 90 in its simplest radical form is \( 3\sqrt{10} \).

The decimal approximation of \( \sqrt{90} \) is approximately 9.486.

The square root of 90, when expressed in simplest radical form, is \( 3\sqrt{10} \).

Here are some key properties:

• Rational vs. Irrational: The square root of 90 is an irrational number because 90 is not a perfect square.
• Decimal Approximation: The decimal approximation of \( \sqrt{90} \) is approximately 9.486832980505138.
• Expression with Exponents: \( \sqrt{90} \) can also be expressed as \( 90^{1/2} \) or \( 10^{3/2} \).

The square root of 90, \( \sqrt{90} \), in its simplest radical form is \( 3\sqrt{10} \). Here are some practical applications and examples:

1. Real-world Problems: Calculating dimensions in geometry where the area is 90 square units and finding the side length of the square or rectangle.
2. Math Problems and Solutions: Solving equations involving geometric shapes or physical quantities where the square root of 90 appears.
3. Interactive Questions: Educational exercises and quizzes that test understanding of simplifying radicals and applying them in different contexts.

• Is the square root of 90 rational?
  No, the square root of 90 is irrational because 90 is not a perfect square.
• How to calculate square root of 90 without a calculator?
  The square root of 90 can be approximated manually by understanding its relationship with perfect squares, such as \( 9^2 = 81 \) and \( 10^2 = 100 \).
• What is the simplified radical form of the square root of 90?
  The simplest radical form of \( \sqrt{90} \) is \( 3\sqrt{10} \).