Square Root of 5 Times by Square Root of 45?

Square roots are an intriguing aspect of mathematics, offering insights into the relationships between numbers and the fundamental principles of multiplication. In this blog post, we will delve into the multiplication of square roots, specifically examining the product of √5 × √45. We’ll explore the underlying concepts, simplify the expression, and discuss the broader significance of square roots in mathematics and real-world applications.

Square Root of 5 Times by Square Root of 45?

The product of the square root of 5 (√5) and the square root of 45 (√45) is approximately equal to 6.70821 multiplied by the square root of 5 (√5).

Understanding Square Roots

Before we embark on our exploration of √5 × √45, let’s refresh our understanding of square roots.

The square root of a number “x,” denoted as √x, is a value that, when multiplied by itself, results in “x.” In other words, (√x) × (√x) = x. For example, the square root of 25 is 5 because 5 × 5 = 25.

Square roots are a crucial part of mathematics and are encountered in various mathematical and scientific contexts.

The Product of Square Roots

Now, let’s focus on the expression √5 × √45. To calculate this product, we can break it down into several steps:

Step 1: Calculate the individual square roots. Step 2: Multiply the results.

Step 1: Calculate the Square Roots

Let’s start with the square roots of the numbers involved:

√5 and √45

The square root of 5 (√5) is an irrational number, meaning it cannot be expressed as a simple fraction or decimal and has an infinite, non-repeating decimal expansion. Its value is approximately 2.23607.

To calculate the square root of 45 (√45), we can break it down into simpler terms. The prime factorization of 45 is 3 × 3 × 5, which allows us to simplify the square root:

√45 = √(3 × 3 × 5)

Using the properties of square roots, we can separate the square root of a product into the product of square roots:

√45 = √(3 × 3 × 5) = (√3 × √3) × √5

Now, we have √45 expressed as (√3 × √3) × √5.

The square root of 3 (√3) is also an irrational number, approximately equal to 1.73205.

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Step 2: Multiply the Results

Now that we have the values for √5 and √45 in terms of their square roots:

√5 ≈ 2.23607 √45 ≈ (√3 × √3) × √5 ≈ (1.73205 × 1.73205) × √5 ≈ 3 × √5

We can proceed to multiply these values:

√5 × (√3 × √3) × √5 = 2.23607 × 3 × √5 = 6.70821 × √5

So, the product of √5 × √45 is approximately equal to 6.70821 × √5.

Practical Applications

The multiplication of square roots has relevance in various fields:

  1. Geometry: Calculations involving areas and side lengths of geometric shapes often require the multiplication of square roots.
  2. Physics: In physics, square roots appear in equations describing physical phenomena, including those related to motion, energy, and waves.
  3. Engineering: Engineers use square roots in various calculations, such as determining stress, strain, and voltage in electrical circuits.
  4. Finance: Square roots play a role in financial mathematics, especially in calculating volatility and risk in financial markets.
  5. Statistics: Square roots are used in statistical analysis, particularly in calculating standard deviations and variances.


In conclusion, the product of √5 × √45 is approximately equal to 6.70821 × √5. This calculation demonstrates the principles of multiplying square roots and highlights the importance of understanding square roots in mathematics and various scientific and engineering disciplines. Square roots are not only essential in theoretical mathematics but also in practical applications, where they enable us to solve complex problems, make accurate measurements, and model natural phenomena. The multiplication of square roots serves as a reminder of the elegance and versatility of mathematical concepts in our everyday lives.

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