3 Square Root of 125: A Comprehensive Guide

The expression "3 square root of 125" can be simplified and understood in a few steps.

Simplification Steps

1. Factorize 125: \( 125 = 5^3 \)
2. Express the square root: \( \sqrt{125} = \sqrt{5^3} = \sqrt{25 \cdot 5} = 5 \sqrt{5} \)
3. Multiply by 3: \( 3 \times 5 \sqrt{5} = 15 \sqrt{5} \)

Therefore, the simplified form of \( 3 \sqrt{125} \) is \( 15 \sqrt{5} \).

Exact and Decimal Forms

• Exact Form: \( 15 \sqrt{5} \)
• Decimal Form: \( 15 \sqrt{5} \approx 33.541 \)

Alternate Interpretations

It's important to note that the cube root of 125 is often confused with the square root. The correct interpretation is:

• Cube Root of 125: \( \sqrt[3]{125} = 5 \)

When dealing with the expression "3 square root of 125," it is essential to understand both the simplification process and its implications in mathematics. This term is a combination of a scalar (3) and a radical expression (square root of 125), often encountered in algebraic operations. Simplifying this expression involves factoring and rationalizing the radicand, providing insights into its exact and decimal forms. Here, we explore step-by-step methods to simplify and understand the value of "3 square root of 125".

1. Identify the components: 3 and √125.
2. Factor the radicand (125) as 5³.
3. Separate the square root: √125 = √(5² * 5) = 5√5.
4. Multiply the scalar: 3 * 5√5 = 15√5.
5. Express the result: 15√5 or approximately 33.541.

Understanding these steps helps in simplifying similar expressions and highlights the importance of prime factorization and multiplication in algebra.

The expression "3 square root of 125" can be simplified through a series of steps involving factorization and multiplication. Here is the detailed process:

1. Factor the radicand: Begin by expressing 125 as a product of its prime factors. We have: \[ 125 = 5^3 \]
2. Rewrite the square root: Next, express the square root of 125 using its prime factors: \[ \sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5} \]
3. Multiply by the scalar: Now, multiply the simplified square root by the scalar (3): \[ 3 \times 5\sqrt{5} = 15\sqrt{5} \]
4. Exact and decimal forms: The result can be expressed in exact form as \(15\sqrt{5}\) and in decimal form as approximately 33.54.

Through these steps, we achieve a simplified form of the expression "3 square root of 125," demonstrating the importance of factorization and multiplication in algebraic simplification.

To simplify \(3 \sqrt{125}\), follow these detailed steps:

1. Factorize the number under the square root:
   • \(125 = 5^3 = 5 \times 25 = 5 \times 5 \times 5\)
2. Express the square root of the product:
   • \(\sqrt{125} = \sqrt{5^3} = \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5} = 5 \sqrt{5}\)
3. Multiply the simplified square root by 3:
   • \(3 \times \sqrt{125} = 3 \times 5 \sqrt{5} = 15 \sqrt{5}\)

Thus, the simplified form of \(3 \sqrt{125}\) is \(15 \sqrt{5}\).

The Prime Factorization Method is a systematic approach to simplifying square roots by expressing the number under the square root as a product of prime factors. Here’s a step-by-step guide to finding the square root of a number using the prime factorization method:

1. Prime Factorization of the Number

   Begin by expressing the given number as a product of its prime factors. For example, for 125, the prime factorization is:

   \[ 125 = 5 \times 5 \times 5 \]

2. Pairing the Prime Factors

   Group the prime factors into pairs. For the number 125, we have:

   \[ 125 = 5^2 \times 5 \]

3. Extracting a Factor from Each Pair

   For each pair of prime factors, take one factor out of the square root. Here, we have one pair of 5’s:

   \[ \sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5} \]

4. Multiplying the Extracted Factors

   Multiply the extracted factors outside the square root by any remaining factors inside. In this case, we end up with:

   \[ 5\sqrt{5} \]

Therefore, the simplified form of \(\sqrt{125}\) is \(5\sqrt{5}\).

Example: Simplifying \(3\sqrt{125}\)

Using the result from our prime factorization, we can further simplify \(3\sqrt{125}\) as follows:

\[ 3\sqrt{125} = 3 \times 5\sqrt{5} = 15\sqrt{5} \]

Thus, the simplified form of \(3\sqrt{125}\) is \(15\sqrt{5}\).

Additional Examples

• Example 1: \( \sqrt{81} \)

  Prime Factorization: \( 81 = 3 \times 3 \times 3 \times 3 \)

  Paired Factors: \( 81 = 3^2 \times 3^2 \)

  Simplified: \( \sqrt{81} = 3 \times 3 = 9 \)

• Example 2: \( \sqrt{100} \)

  Prime Factorization: \( 100 = 2 \times 2 \times 5 \times 5 \)

  Paired Factors: \( 100 = 2^2 \times 5^2 \)

  Simplified: \( \sqrt{100} = 2 \times 5 = 10 \)

The Prime Factorization Method provides a clear and methodical approach to simplifying square roots, making it easier to work with radical expressions in their simplest form.

The expression "3 square root of 125" is related to several important mathematical concepts. Here we will explore the connections to prime factorization, irrational numbers, geometric applications, and more.

Prime Factorization

Prime factorization is essential in simplifying the square root of non-perfect square numbers. For 125, the prime factors are:

• 125 = 5 × 5 × 5 = 5³
• Thus, √125 = √(5³) = 5√5

Irrational Numbers

The simplified form 5√5 demonstrates that the square root of 125 is an irrational number. An irrational number cannot be expressed as a simple fraction, and its decimal form is non-repeating and non-terminating.

Geometric Applications

Square roots are used in geometry, particularly in the Pythagorean theorem, which relates the lengths of the sides of a right triangle. For instance, if a square has an area of 125 square units, the length of each side is √125, or approximately 11.18 units.

Cube Root of 125

While the square root of 125 is important, its cube root is also noteworthy:

• 125 = 5 × 5 × 5
• ∛125 = 5

This illustrates how different root operations can provide insights into the structure of numbers.

Properties of Square Roots

Understanding properties of square roots aids in various mathematical operations:

• The square root of a product: √(a × b) = √a × √b
• Simplifying radicals: Taking pairs of prime factors out of the square root

Real-World Applications

Simplifying square roots is crucial in numerous fields:

• Physics: Calculating forces, velocities, and energy levels often involves square roots.
• Engineering: Determining dimensions and tolerances in design and manufacturing.
• Architecture: Calculating areas, volumes, and other geometric properties in construction.

Comparison with Other Roots

Comparing square roots with cube roots and other roots helps in understanding their differences and applications. For example:

• The square root of 125 simplifies to 5√5.
• The cube root of 125 is a simple integer, 5.

Common Misconceptions

Addressing common misconceptions enhances comprehension:

• Misconception: The square root of 125 is often incorrectly simplified as a whole number.
• Correction: The correct simplification is 5√5, which is an irrational number.

The expression \( 3\sqrt{125} \) finds its use in various practical scenarios. Here are some detailed examples and applications:

• Calculating the Side Length of a Square Pool:

  Suppose you have a square pool with an area of 125 square feet, and you want to determine the side length of the pool. The side length can be found by taking the square root of the area.

  1. Area of the pool, \( A = 125 \, \text{sq ft} \)
  2. Side length, \( s = \sqrt{125} \)
  3. Simplifying, \( \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \)
  4. The side length of the pool is approximately \( 5\sqrt{5} \approx 11.18 \, \text{ft} \)

• Comparison with Other Roots:

  Understanding the difference between various roots helps in comparing different quantities.

  Expression	Exact Form	Decimal Approximation
  \( 3\sqrt{125} \)	\( 15\sqrt{5} \)	\( \approx 33.54 \)
  \( \sqrt{125} \)	\( 5\sqrt{5} \)	\( \approx 11.18 \)
  \( \sqrt{80} \)	\( 4\sqrt{5} \)	\( \approx 8.94 \)
  \( \sqrt{405} \)	\( 9\sqrt{5} \)	\( \approx 20.12 \)

• Applications in Geometry and Construction:

  Square roots are often used in geometry and construction to determine dimensions, areas, and other measurements.

  1. Finding the length of the diagonal in a square: Given a square with side length \( a \), the diagonal length \( d \) is \( d = a\sqrt{2} \).
  2. Scaling objects: When scaling objects, the dimensions are often multiplied by a square root to maintain proportions.

• What is the exact value of 3√125?

  The exact value of \(3\sqrt{125}\) is \(15\sqrt{5}\). This is found by simplifying the square root of 125 as \(5\sqrt{5}\), and then multiplying by 3.

• How is the square root of 125 simplified?

  The square root of 125 is simplified by expressing 125 as a product of its prime factors: \(125 = 5^3\). Then, the square root of 125 is simplified to \(5\sqrt{5}\).

• Is 125 a perfect square or a cube number?

  125 is not a perfect square, but it is a perfect cube. Specifically, \(125 = 5^3\), so its cube root is 5.

• What is the decimal approximation of \(3\sqrt{125}\)?

  The decimal approximation of \(3\sqrt{125}\) is approximately 33.541. This is calculated using \(3 \times 11.1803398875\), where 11.1803398875 is the decimal form of \(\sqrt{125}\).

• Is the square root of 125 a rational or irrational number?

  The square root of 125 is an irrational number. This is because its decimal form is non-terminating and non-repeating, which means it cannot be expressed as a ratio of two integers.

• What methods can be used to find the square root of 125?

  Two common methods to find the square root of 125 are the prime factorization method and the long division method. Prime factorization simplifies 125 into \(5^2 \times 5\), resulting in \(5\sqrt{5}\). The long division method provides a step-by-step process to approximate the square root.