Understanding the 3 Square Root of 125: A Complete Guide

To calculate the expression \(3 \sqrt{125}\), follow these steps:

Step-by-Step Calculation

1. First, find the square root of 125.
2. The square root of 125 can be simplified as follows:

   \[
   \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5 \sqrt{5}
   \]

3. Next, multiply the result by 3:

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

Conclusion

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

Decimal Approximation

For practical purposes, it may be useful to approximate the value of \(15 \sqrt{5}\) as a decimal:

\[
15 \sqrt{5} \approx 15 \times 2.236 = 33.54
\]

Therefore, \(3 \sqrt{125} \approx 33.54\).

The concept of square roots is fundamental in mathematics, providing a basis for understanding more complex algebraic and geometric principles. A square root of a number x is a number y such that y² = x. In other words, it is a value that, when multiplied by itself, gives the original number.

Square roots are denoted by the radical symbol (√). For example, the square root of 9 is written as √9, and since 3 * 3 = 9, √9 = 3. Square roots can be classified into two categories:

• Perfect Squares: Numbers that have integer square roots. Examples include 1, 4, 9, 16, and 25.
• Non-Perfect Squares: Numbers that do not have integer square roots, resulting in irrational numbers. Examples include 2, 3, 5, and 10.

To understand square roots better, it’s helpful to explore some basic properties:

• The square root of 0 is 0: √0 = 0.
• The square root of 1 is 1: √1 = 1.
• Square roots are non-negative: For any non-negative number x, √x ≥ 0.

Square roots also follow specific rules when it comes to multiplication and division:

• Multiplication: √(a * b) = √a * √b. For instance, √(4 * 9) = √4 * √9 = 2 * 3 = 6.
• Division: √(a / b) = √a / √b. For example, √(16 / 4) = √16 / √4 = 4 / 2 = 2.

When dealing with square roots, it's essential to simplify the expressions as much as possible. Simplification involves breaking down the number inside the radical, known as the radicand, into its prime factors. This process makes it easier to extract square roots, especially for larger numbers.

Let's take an example to illustrate the concept:

Consider the number 125. To find its square root, we first break it down into prime factors:

1. 125 = 5 * 25
2. 25 = 5 * 5

So, 125 = 5 * 5 * 5, or 5³. Since the square root of 125 is not an integer, it is an irrational number. The simplified form of the square root of 125 is:

√125 = √(5³) = 5√5

Understanding these foundational aspects of square roots will make it easier to tackle more advanced topics and apply these concepts in various mathematical problems.

The expression \(3\sqrt{125}\) involves both multiplication and the operation of taking a square root. To understand this concept better, let's break it down step by step:

1. Factor the Radicand: The radicand in our expression is 125. We start by breaking it down into its prime factors: \[ 125 = 5 \times 25 = 5 \times 5 \times 5 = 5^3 \]
2. Rewrite Under the Radical: Since 125 can be expressed as \(5^3\), we can rewrite the square root of 125 as: \[ \sqrt{125} = \sqrt{5^2 \times 5} = \sqrt{25 \times 5} \]
3. Simplify the Radical: We know that \(\sqrt{25} = 5\), so we can simplify the expression under the radical: \[ \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5} \]
4. Multiply by the Coefficient: Now, we multiply the simplified radical by 3: \[ 3\sqrt{125} = 3 \times 5\sqrt{5} = 15\sqrt{5} \]

Therefore, the expression \(3\sqrt{125}\) simplifies to \(15\sqrt{5}\). This shows the power of breaking down the radicand into its prime factors and simplifying the radical before multiplying by any coefficients.

To provide more context, here are some forms of the result:

• Exact Form: \(15\sqrt{5}\)
• Decimal Form: Using a calculator, \( \sqrt{5} \approx 2.236\), so: \[ 15\sqrt{5} \approx 15 \times 2.236 = 33.54 \]

Understanding the concept of the square root and how to manipulate it through factorization and simplification is crucial in algebra. By breaking down the radicand and using properties of radicals, we can simplify complex expressions effectively.

To understand the concept of the 3 square root of 125, we first need to break down the radicand, which is the number under the radical sign.

The number 125 can be factored into its prime components. The process is as follows:

1. Start with the number 125.
2. Determine the prime factors of 125. Since 125 is an odd number and ends in 5, it is divisible by 5.
3. Divide 125 by 5:

   \[
   125 \div 5 = 25
   \]

4. Next, factor 25. It is also divisible by 5:

   \[
   25 \div 5 = 5
   \]

5. Finally, factor 5. Since 5 is a prime number, it cannot be divided further:

   \[
   5 \div 5 = 1
   \]

Putting it all together, we get:

\[
125 = 5 \times 5 \times 5 = 5^3
\]

Thus, the prime factorization of 125 is \( 5^3 \). This breakdown helps us simplify the calculation of the 3 square root of 125.

To find the 3 square root of 125, we recognize that we are looking for:

\[
3 \sqrt{125} = 3 \sqrt{5^3}
\]

Using the properties of exponents and radicals, we can simplify this expression further:

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

This simplification can also be represented as:

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

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

Understanding this process is crucial as it lays the foundation for more complex operations involving radicals and helps in simplifying radical expressions effectively.

The process of simplifying the expression \(3 \sqrt{125}\) involves several steps. Here, we will break it down step by step to understand how to achieve the simplest form.

1. Understand the given expression: The expression we need to simplify is \(3 \sqrt{125}\).
2. Prime Factorization: First, we find the prime factorization of 125.
   • 125 can be divided by 5: \(125 \div 5 = 25\)
   • 25 can also be divided by 5: \(25 \div 5 = 5\)
   • Finally, 5 is a prime number: \(5 \div 5 = 1\)
   Therefore, the prime factorization of 125 is \(5 \times 5 \times 5 = 5^3\).
3. Rewrite the square root: Using the prime factorization, we rewrite the square root of 125.

   \[
   \sqrt{125} = \sqrt{5^3} = \sqrt{5^2 \times 5}
   \]

4. Extract the square root: We know that the square root of a square number is simply the base of the exponent.

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

5. Multiply by 3: Now, we multiply the simplified square root by 3.

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

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

When working with square roots, it's useful to express the result in both exact and decimal forms. Here, we will look at the exact and decimal forms of \(3 \sqrt{125}\).

1. Exact Form: The exact form of \(3 \sqrt{125}\) is derived by simplifying the square root as we did earlier.
   • Recall that \( \sqrt{125} = 5 \sqrt{5} \).
   • Therefore, \(3 \sqrt{125} = 3 \times 5 \sqrt{5} = 15 \sqrt{5}\).

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

2. Decimal Form: To find the decimal form, we first need to approximate the value of \( \sqrt{5} \).
   • The value of \( \sqrt{5} \approx 2.236 \).
   • Now, multiply this value by 15:

     \[
     15 \times 2.236 \approx 33.54
     \]

   Thus, the decimal form of \(3 \sqrt{125}\) is approximately 33.54.

In summary:

• Exact Form: \(15 \sqrt{5}\)
• Decimal Form: Approximately 33.54

The prime factorization method is a useful technique for simplifying square roots. To understand this method, let's break down the process of finding the prime factors of the radicand and then simplifying the expression.

1. Step 1: Find the Prime Factors of 125

   First, we need to find the prime factors of the number 125. We start by dividing the number by the smallest prime number, which is 2. Since 125 is not divisible by 2, we move to the next prime number, which is 3. Again, 125 is not divisible by 3. The next prime number is 5.

   125 is divisible by 5:

   \[
   125 \div 5 = 25
   \]

   We continue factoring 25 by 5:

   \[
   25 \div 5 = 5
   \]

   And finally:

   \[
   5 \div 5 = 1
   \]

   So, the prime factorization of 125 is:

   \[
   125 = 5 \times 5 \times 5 = 5^3
   \]

2. Step 2: Apply the Prime Factors to the Radical

   Next, we use the prime factors to simplify the square root. Since we are dealing with the 3rd square root (cube root) of 125, we write:

   \[
   \sqrt[3]{125} = \sqrt[3]{5^3}
   \]

3. Step 3: Simplify the Expression

   To simplify the expression, we use the property of radicals that states \(\sqrt[n]{a^n} = a\). Therefore:

   \[
   \sqrt[3]{5^3} = 5
   \]

   So, the simplified form of the 3rd square root of 125 is 5.

By using the prime factorization method, we can easily simplify the 3rd square root of 125 to its simplest form.

Multiplication and division of radicals follow specific rules that make these operations straightforward once understood. Here, we will explore these rules step-by-step using the example of the 3rd square root (cube root) of 125.

1. Multiplication of Radicals

   To multiply radicals with the same index, we can use the property:

   \[
   \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b}
   \]

   For example, let's multiply \(\sqrt[3]{125}\) with \(\sqrt[3]{8}\):

   \[
   \sqrt[3]{125} \times \sqrt[3]{8} = \sqrt[3]{125 \times 8}
   \]

   Since \(125 = 5^3\) and \(8 = 2^3\), we can simplify the multiplication:

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

2. Division of Radicals

   To divide radicals with the same index, we use the property:

   \[
   \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}
   \]

   For instance, let's divide \(\sqrt[3]{125}\) by \(\sqrt[3]{27}\):

   \[
   \frac{\sqrt[3]{125}}{\sqrt[3]{27}} = \sqrt[3]{\frac{125}{27}}
   \]

   We can simplify the fraction inside the radical. Since \(125 = 5^3\) and \(27 = 3^3\), we get:

   \[
   \sqrt[3]{\frac{5^3}{3^3}} = \sqrt[3]{\left(\frac{5}{3}\right)^3} = \frac{5}{3}
   \]

These examples demonstrate how the properties of radicals can be applied to simplify both multiplication and division of radicals, making it easier to handle these operations in mathematical problems.

Radicals, including the 3rd square root (cube root), have several properties that simplify their manipulation and calculation. These properties are fundamental to working with radicals in algebra and other areas of mathematics. Here, we will explore these properties in detail.

1. Product Property of Radicals

   The product property of radicals states that the radical of a product is the product of the radicals:

   \[
   \sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}
   \]

   For example, for the cube roots of 8 and 27:

   \[
   \sqrt[3]{8 \times 27} = \sqrt[3]{8} \times \sqrt[3]{27}
   \]

   Since \(8 = 2^3\) and \(27 = 3^3\), we get:

   \[
   \sqrt[3]{8} = 2 \quad \text{and} \quad \sqrt[3]{27} = 3
   \]

   Thus:

   \[
   \sqrt[3]{8 \times 27} = 2 \times 3 = 6
   \]

2. Quotient Property of Radicals

   The quotient property of radicals states that the radical of a quotient is the quotient of the radicals:

   \[
   \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
   \]

   For example, for the cube roots of 125 and 27:

   \[
   \sqrt[3]{\frac{125}{27}} = \frac{\sqrt[3]{125}}{\sqrt[3]{27}}
   \]

   Since \(125 = 5^3\) and \(27 = 3^3\), we get:

   \[
   \sqrt[3]{125} = 5 \quad \text{and} \quad \sqrt[3]{27} = 3
   \]

   Thus:

   \[
   \sqrt[3]{\frac{125}{27}} = \frac{5}{3}
   \]

3. Power Property of Radicals

   The power property of radicals states that a radical can be expressed as a power:

   \[
   \sqrt[n]{a} = a^{\frac{1}{n}}
   \]

   For example, the cube root of 125 can be written as:

   \[
   \sqrt[3]{125} = 125^{\frac{1}{3}}
   \]

   Using the property that \(125 = 5^3\), we get:

   \[
   125^{\frac{1}{3}} = (5^3)^{\frac{1}{3}} = 5
   \]

Understanding these properties of radicals allows for more effective and efficient problem solving when dealing with radical expressions in mathematics.

Understanding the 3rd root (or cube root) of 125 can be challenging, and there are several common mistakes and misconceptions that students often encounter. Here are some of the key points to keep in mind to avoid these errors:

• Confusing Square Root with Cube Root:

  One of the most common errors is confusing the square root with the cube root. The square root of a number is a value that, when multiplied by itself, gives the original number. The cube root is a value that, when multiplied by itself twice, gives the original number. For instance, the square root of 9 is 3 (because 3 x 3 = 9), while the cube root of 27 is 3 (because 3 x 3 x 3 = 27). Make sure to distinguish between these operations.

• Incorrectly Simplifying the Expression:

  Another common mistake is incorrectly simplifying the cube root of 125. The correct simplification involves recognizing that 125 is a perfect cube. Since \( 125 = 5^3 \), the cube root of 125 is 5. Incorrect simplifications might involve incorrect factorization or arithmetic errors.

• Misunderstanding the Properties of Radicals:

  It's crucial to apply the properties of radicals correctly. For example, some students might incorrectly assume that the cube root of a product is the product of the cube roots, which is true (i.e., \( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \)). However, this must be applied correctly to avoid mistakes.

• Overlooking Negative Roots:

  When dealing with cube roots, it's important to remember that they can yield negative results for negative numbers. For example, \( \sqrt[3]{-125} = -5 \). This is different from square roots, which do not have real negative results for real numbers.

• Forgetting the Radical Properties in Multiplication and Division:

  Another common mistake is forgetting how to handle radicals when multiplying or dividing. For instance, the product rule \( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \) and the quotient rule \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \) need to be applied correctly to simplify expressions involving cube roots.

By being aware of these common mistakes, you can approach problems involving cube roots with greater confidence and accuracy. Practice these concepts regularly to reinforce your understanding and avoid these pitfalls.

The concept of the cube root, or the 3rd root, is crucial in various mathematical contexts. Understanding the cube root of numbers like 125 is not only fundamental in algebra but also has practical applications in different fields. Here are some of the key areas where the cube root of 125 and cube roots, in general, play a significant role:

• Solving Polynomial Equations:

  Cubic equations, which are polynomial equations of degree three, often require the use of cube roots to find their solutions. For instance, solving the equation \( x^3 = 125 \) involves finding the cube root of 125, which is 5, giving us \( x = 5 \). This principle is used extensively in algebra.

• Geometry and Volume Calculations:

  Cube roots are essential in geometry, especially when dealing with volumes. For example, the volume of a cube is calculated as \( V = s^3 \), where \( s \) is the side length. To find the side length when the volume is given, we use the cube root. If the volume is 125 cubic units, the side length is \( \sqrt[3]{125} = 5 \) units.

• Scalability and Proportionality:

  In fields like physics and engineering, cube roots help understand relationships involving scale. For instance, when scaling up a three-dimensional object, the volume scales with the cube of the scaling factor. If a model is scaled up by a factor of 5, its volume becomes \( 5^3 = 125 \) times larger, and understanding cube roots helps in reverse-engineering the original dimensions from the scaled version.

• Exponential Growth and Decay:

  Cube roots also appear in the study of exponential growth and decay, particularly in contexts where quantities grow or shrink by a factor that involves three-dimensional volume changes. For example, in certain biological growth models, the size of an organism might increase by the cube of its age, and understanding cube roots can help analyze these relationships.

• Physics and Dynamics:

  In physics, cube roots are used in various calculations, such as in understanding the relationship between the mass and the radius of spherical objects. For example, if the mass of a planet is 125 times that of a smaller one, the radius (assuming density is constant) can be found using the cube root, leading to \( \sqrt[3]{125} = 5 \). This principle is used in astrophysics and planetary science.

Understanding cube roots like the 3rd root of 125 is not only academically important but also provides a foundation for practical problem-solving across various scientific and engineering disciplines.

To solidify your understanding of cube roots and specifically the cube root of 125, try solving the following practice problems. These problems will help you apply the concept in various scenarios and ensure you are comfortable with the process.

1. Basic Cube Root Calculation:

   Find the cube root of the following numbers:

   • \( \sqrt[3]{27} \)
   • \( \sqrt[3]{64} \)
   • \( \sqrt[3]{343} \)

   Solution:

   • \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \)
   • \( \sqrt[3]{64} = 4 \) because \( 4 \times 4 \times 4 = 64 \)
   • \( \sqrt[3]{343} = 7 \) because \( 7 \times 7 \times 7 = 343 \)
2. Applying Cube Roots to Volume Problems:

   If the volume of a cube is 125 cubic units, find the length of one side of the cube.

   Solution:

   The side length \( s \) can be found by taking the cube root of the volume:

   \( s = \sqrt[3]{125} = 5 \) units

3. Solving for Unknowns Using Cube Roots:

   Solve for \( x \) in the equation \( x^3 = 125 \).

   Solution:

   To find \( x \), take the cube root of both sides:

   \( x = \sqrt[3]{125} = 5 \)

4. Cube Roots of Negative Numbers:

   Find the cube root of the following negative numbers:

   • \( \sqrt[3]{-8} \)
   • \( \sqrt[3]{-27} \)
   • \( \sqrt[3]{-125} \)

   Solution:

   • \( \sqrt[3]{-8} = -2 \) because \( -2 \times -2 \times -2 = -8 \)
   • \( \sqrt[3]{-27} = -3 \) because \( -3 \times -3 \times -3 = -27 \)
   • \( \sqrt[3]{-125} = -5 \) because \( -5 \times -5 \times -5 = -125 \)
5. Combining Cube Roots:

   Simplify the expression \( \sqrt[3]{125 \times 8} \).

   Solution:

   First, find the cube roots of each factor and then multiply them:

   \( \sqrt[3]{125} = 5 \) and \( \sqrt[3]{8} = 2 \)

   Thus, \( \sqrt[3]{125 \times 8} = \sqrt[3]{1000} = \sqrt[3]{10^3} = 10 \)

These practice problems will help reinforce your understanding of cube roots and their applications. Continue practicing with more complex problems to deepen your comprehension.