Simplify Square Root of 128: Easy Step-by-Step Guide

The process of simplifying the square root of 128 involves breaking down the number into its prime factors and simplifying the radical expression. Here is a detailed step-by-step explanation:

Step-by-Step Simplification

1. First, find the prime factorization of 128:
   • 128 is an even number, so it can be divided by 2:
   • 128 ÷ 2 = 64
   • 64 is also even, so continue dividing by 2:
   • 64 ÷ 2 = 32
   • 32 ÷ 2 = 16
   • 16 ÷ 2 = 8
   • 8 ÷ 2 = 4
   • 4 ÷ 2 = 2
2. So, the prime factorization of 128 is:

   \( 128 = 2^7 \)

3. Rewrite the square root of 128 using its prime factorization:

   \( \sqrt{128} = \sqrt{2^7} \)

4. Group the prime factors into pairs:

   \( \sqrt{2^7} = \sqrt{(2^2 \cdot 2^2 \cdot 2^2) \cdot 2} \)

5. Since the square root of a pair of identical factors is just one of those factors, simplify the expression:

   \( \sqrt{(2^2 \cdot 2^2 \cdot 2^2) \cdot 2} = 2^3 \cdot \sqrt{2} \)

6. Calculate the simplified result:

   \( 2^3 = 8 \)

   Thus, \( \sqrt{128} = 8\sqrt{2} \)

Conclusion

The simplified form of the square root of 128 is:

The concept of square roots is fundamental in mathematics, particularly in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for the square root is √.

For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. Mathematically, this is expressed as:

$$

16

=
4

Square roots can be either rational or irrational. A rational square root is an integer or a fraction, such as √25 = 5. An irrational square root is a non-repeating, non-terminating decimal, like √2.

The square root of a number can often be simplified by factoring the number into its prime factors and then simplifying the radical. For example, to simplify the square root of 128:

• First, find the prime factorization of 128, which is 2 × 2 × 2 × 2 × 2 × 2 × 2 (or 2⁷).
• Next, group the factors into pairs: (2 × 2) × (2 × 2) × (2 × 2) × 2.
• Simplify each pair: √(2 × 2) = 2, so you have 2 × 2 × 2 × √2 = 8√2.

Therefore, the simplified form of √128 is 8√2.

Understanding how to simplify square roots is crucial for solving various mathematical problems, including equations and real-world applications where square roots appear, such as in geometry and physics.

A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, if x is the square root of y, then x × x = y or x² = y. The square root is denoted by the radical symbol √, followed by the number. For example, the square root of 128 is written as √128.

Understanding square roots is crucial in various areas of mathematics, including algebra, geometry, and calculus. Here's a step-by-step explanation to understand the concept better:

• Perfect Squares: Numbers like 1, 4, 9, 16, 25, etc., are perfect squares because they are squares of integers (1², 2², 3², etc.).
• Non-Perfect Squares: Numbers that are not perfect squares do not have integers as their square roots. For example, 128 is not a perfect square.
• Simplifying Square Roots: To simplify a square root, factor the number into its prime factors and pair the identical factors. For instance, √128 can be simplified by expressing 128 as a product of its prime factors.

Let's look at an example of simplifying the square root of 128:

1. Factorize 128: 128 = 2 × 64.
2. Recognize that 64 is a perfect square (8²): 128 = 2 × 8².
3. Apply the square root to both factors: √128 = √(8² × 2) = 8√2.

Therefore, the simplified form of √128 is 8√2, which is approximately 11.3137.

Understanding the principles of square roots and their simplification helps in solving more complex mathematical problems and equations, making this a foundational concept in mathematics.

The prime factorization method is an effective way to simplify square roots by breaking down the number into its prime factors. Here, we will simplify the square root of 128 using this method.

1. Find the Prime Factors of 128:

   We start by dividing 128 by the smallest prime number, which is 2, and continue dividing by 2 until we cannot divide evenly anymore.

   128 ÷ 2 = 64

   64 ÷ 2 = 32

   32 ÷ 2 = 16

   16 ÷ 2 = 8

   8 ÷ 2 = 4

   4 ÷ 2 = 2

   2 ÷ 2 = 1

   Thus, the prime factorization of 128 is: \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7 \)

2. Group the Prime Factors:

   To simplify the square root, we pair the prime factors into groups of two:

   \( 128 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2 \)

3. Simplify the Square Root:

   We take one factor from each pair out of the square root:

   \( \sqrt{128} = \sqrt{(2^2) \times (2^2) \times (2^2) \times 2} \)

   \( \sqrt{128} = 2 \times 2 \times 2 \times \sqrt{2} \)

   \( \sqrt{128} = 8\sqrt{2} \)

Thus, the simplified form of \( \sqrt{128} \) is \( 8\sqrt{2} \).

In decimal form, this is approximately 11.3137.

Simplifying the square root of 128 involves breaking down the number into its prime factors and applying the properties of square roots. Here are the detailed steps:

1. Factorize 128: Start by finding the prime factors of 128.

   128 can be factorized as:

   \[ 128 = 2 \times 64 \]

   Then, break down 64:

   \[ 64 = 2 \times 32 \]

   Continue factorizing:

   \[ 32 = 2 \times 16 \]

   \[ 16 = 2 \times 8 \]

   \[ 8 = 2 \times 4 \]

   \[ 4 = 2 \times 2 \]

   Thus, the prime factors of 128 are:

   \[ 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7 \]

2. Group the Prime Factors: Group the factors into pairs of two since we are dealing with square roots.

   \[ 2^7 = (2^2 \times 2^2 \times 2^2) \times 2 = (2 \times 2 \times 2) \times \sqrt{2} = 8 \times \sqrt{2} \]

3. Apply the Square Root Property: Use the property of square roots that states:

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

   Therefore:

   \[ \sqrt{128} = \sqrt{2^7} = \sqrt{(2^2 \times 2^2 \times 2^2) \times 2} = \sqrt{2^2} \times \sqrt{2^2} \times \sqrt{2^2} \times \sqrt{2} \]

   Simplify the pairs:

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

   So we get:

   \[ 2 \times 2 \times 2 \times \sqrt{2} = 8 \sqrt{2} \]

4. Result: Combine the simplified factors to get the final answer:

   \[ \sqrt{128} = 8 \sqrt{2} \]

Thus, the simplest form of the square root of 128 is:

\[ 8 \sqrt{2} \]

To simplify the square root of 128, we use the method of prime factorization. Here is a step-by-step detailed example:

1. First, find the prime factorization of 128:

   • 128 can be factored into \(2 \times 64\).
   • 64 can be further factored into \(2 \times 32\).
   • Continue factoring: \(32 = 2 \times 16\), \(16 = 2 \times 8\), \(8 = 2 \times 4\), and \(4 = 2 \times 2\).
   • So, \(128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\), or \(2^7\).

2. Group the prime factors into pairs:

   • Since we have \(2^7\), we can group them as \((2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2\).

3. Simplify by taking one number from each pair out of the square root:

   • \(\sqrt{128} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2}\).
   • This simplifies to \(\sqrt{(2^2) \times (2^2) \times (2^2) \times 2} = 2 \times 2 \times 2 \times \sqrt{2}\).

4. Calculate the simplified form:

   • \(2 \times 2 \times 2 = 8\), so \(\sqrt{128} = 8\sqrt{2}\).

Thus, the simplified form of \(\sqrt{128}\) is \(8\sqrt{2}\).

Simplifying radicals involves finding the simplest form of a square root. Here are several alternative methods to simplify radicals:

Method 1: Using the Product Rule

The product rule allows us to simplify the square root of a product of numbers:

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

Steps to use the product rule:

1. Factor the radicand (the number inside the square root) into its prime factors.
2. Group the factors into pairs of perfect squares.
3. Apply the product rule to simplify.

Example:

\[\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\]

Method 2: Using the Quotient Rule

The quotient rule helps simplify the square root of a fraction:

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

Steps to use the quotient rule:

1. Write the square root of the fraction as the quotient of the square roots of the numerator and denominator.
2. Simplify the square roots separately.
3. Combine the simplified square roots.

Example:

\[\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}\]

Method 3: Rationalizing the Denominator

Rationalizing the denominator involves eliminating the square root from the denominator of a fraction:

Steps to rationalize the denominator:

1. Multiply both the numerator and the denominator by the square root that is in the denominator.
2. Simplify the resulting expression.

Example:

\[\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]

Method 4: Simplifying Square Roots of Negative Numbers

When dealing with negative numbers under the square root, use the imaginary unit \(i\):

\[i = \sqrt{-1}\]

Steps to simplify square roots of negative numbers:

1. Factor out the negative sign and use \(i\).
2. Simplify the remaining square root as usual.

Example:

\[\sqrt{-18} = \sqrt{-1 \times 18} = i\sqrt{18} = 3i\sqrt{2}\]

These methods provide alternative approaches to simplifying square roots, each useful depending on the specific form of the radical expression.

When simplifying square roots, particularly for numbers like 128, several common mistakes can occur. Understanding these mistakes can help ensure accuracy in your calculations. Below are some frequent errors and tips on how to avoid them:

• Incorrect Prime Factorization:

  One of the most common mistakes is incorrect prime factorization. Ensure each step of breaking down the number into its prime factors is done correctly. For 128, the prime factorization is \(128 = 2^7\). Misidentifying a factor can lead to incorrect simplification.

• Improper Grouping of Factors:

  Another frequent error is improper grouping of prime factors. Remember that pairs of prime factors should be grouped together. For instance, in \(128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\), there are three pairs of 2s, and one unpaired 2.

• Ignoring the Remaining Factors:

  Sometimes, students forget to include the remaining unpaired factor under the radical. After grouping \(128 = 2^7\), we have \((2^3) \times \sqrt{2} = 8\sqrt{2}\). Ensure all factors are accounted for.

• Forgetting to Simplify Completely:

  Ensure the expression is simplified as much as possible. For example, simplifying \(\sqrt{128}\) completely yields \(8\sqrt{2}\), not just \(8\sqrt{2}\).

• Mathematical Errors:

  Simple arithmetic mistakes can occur during calculations. Double-check each step of your work, especially during multiplication and addition, to avoid errors.

• Incorrect Application of Square Root Properties:

  Misapplying properties of square roots can lead to incorrect answers. Remember that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) and use these properties correctly.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy in simplifying square roots.

Simplified square roots have a wide range of applications in various fields. Understanding these applications can help appreciate the practical importance of simplifying radicals. Below are some notable examples:

• Finance:

  In finance, simplified square roots are used to calculate the volatility of stocks. The standard deviation, which is the square root of the variance, helps investors assess the risk associated with stock price fluctuations.

• Architecture and Engineering:

  Simplified square roots play a crucial role in determining the natural frequencies of structures like bridges and buildings. This helps engineers predict how these structures will respond to different loads, such as wind or traffic.

• Science:

  In scientific calculations, square roots are used to determine various physical properties, such as the velocity of moving objects, the intensity of sound waves, and the amount of radiation absorbed by materials.

• Statistics:

  In statistics, square roots are used to calculate the standard deviation, which measures the dispersion of a data set. This helps in understanding the variability and reliability of data.

• Geometry:

  Geometry uses square roots to solve problems involving right triangles, such as in the Pythagorean theorem, and to compute areas and perimeters of various shapes.

• Computer Science:

  In computer science, square roots are used in algorithms for encryption, image processing, and game physics, helping to secure data and create realistic simulations.

• Navigation:

  In navigation, square roots are used to calculate distances between points on a map or globe, helping pilots and sailors determine the most efficient routes.

• Electrical Engineering:

  Electrical engineers use square roots to calculate power, voltage, and current in circuits, which are fundamental in designing and analyzing electrical systems.

• Photography:

  In photography, the f-number of a camera lens, which affects the amount of light entering the camera, is related to the square root of the aperture area. Adjusting the f-number changes the exposure and depth of field in photos.

• Telecommunication:

  In telecommunications, the inverse square law, which describes how signal strength decreases with distance, involves the use of square roots to ensure proper signal transmission and reception.

Below are some practice problems to help you master the simplification of square roots, including the square root of 128.

1. Simplify \(\sqrt{72}\)
2. Simplify \(\sqrt{200}\)
3. Simplify \(\sqrt{50}\)
4. Simplify \(\sqrt{18}\)
5. Simplify \(\sqrt{450}\)
6. Simplify \(\sqrt{32}\)
7. Simplify \(\sqrt{98}\)
8. Simplify \(\sqrt{242}\)
9. Simplify \(\sqrt{500}\)
10. Simplify \(\sqrt{128}\)

To help guide you, let's simplify the first problem together:

1. \(\sqrt{72}\):
   • Step 1: Find the prime factorization of 72: \(72 = 2^3 \times 3^2\)
   • Step 2: Rewrite under the square root: \(\sqrt{72} = \sqrt{2^3 \times 3^2}\)
   • Step 3: Simplify by taking the square root of each factor: \(\sqrt{72} = \sqrt{(2^2 \times 2) \times 3^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}\)

Try to use the same steps to solve the other problems.