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

The square root of 128 can be simplified by breaking it down into its prime factors.

Step-by-Step Process

1. Identify the prime factors of 128.
2. Express 128 as a product of these prime factors.
3. Simplify the square root using these factors.

Step 1: Prime Factorization of 128

128 can be factorized as follows:

128 = 2 × 64

64 = 2 × 32

32 = 2 × 16

16 = 2 × 8

8 = 2 × 4

4 = 2 × 2

So, 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷

Step 2: Simplifying the Square Root

We can write the square root of 128 as:

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

Since 2⁷ = 2⁶ × 2, we have:

\(\sqrt{2^7} = \sqrt{2^6 \times 2} = \sqrt{2^6} \times \sqrt{2} = 2^3 \times \sqrt{2}\)

Final Simplified Form

Therefore, the simplified form of the square root of 128 is:

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

Simplifying square roots is a fundamental concept in algebra that helps in making expressions involving square roots easier to understand and work with. When we simplify a square root, we aim to express it in the simplest possible radical form.

To simplify a square root, follow these basic steps:

• Find the prime factorization of the number under the square root.
• Group the prime factors into pairs.
• Move each pair of prime factors out from under the square root as a single factor.

Here is a step-by-step process to simplify square roots using the example of √128:

1. Find the prime factorization of 128:

   128 can be broken down into prime factors as follows:

   • 128 ÷ 2 = 64
   • 64 ÷ 2 = 32
   • 32 ÷ 2 = 16
   • 16 ÷ 2 = 8
   • 8 ÷ 2 = 4
   • 4 ÷ 2 = 2
   • 2 ÷ 2 = 1

   So, the prime factorization of 128 is 2 × 2 × 2 × 2 × 2 × 2 × 2, or 2⁷.

2. Group the prime factors into pairs:

   Since we have 7 factors of 2, we can group them as (2 × 2) × (2 × 2) × (2 × 2) × 2.

3. Move each pair of prime factors out from under the square root:

   Each pair of 2's can be moved out as a single 2. So, we get:

   √(2 × 2 × 2 × 2 × 2 × 2 × 2) = 2 × 2 × 2 × √2 = 8√2

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

By understanding and applying these steps, you can simplify any square root, making it easier to work with in algebraic expressions.

Square roots are mathematical operations that undo the process of squaring a number. If you square a number, you multiply it by itself. For example, 3 squared (written as \(3^2\)) is 9. The square root of 9 (written as \(\sqrt{9}\)) is 3. Thus, the square root of a number is a value that, when multiplied by itself, gives the original number.

Square roots can be represented using the radical symbol \( \sqrt{} \). For instance, the square root of 25 is written as \( \sqrt{25} \) and equals 5 because \(5^2 = 25\).

It's important to note the properties of square roots:

• For any positive real number \( a \): \( \sqrt{a^2} = a \) and \( (\sqrt{a})^2 = a \).
• The product property of square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• The quotient property of square roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Square roots are used extensively in algebra, geometry, and various fields of science and engineering. They help solve equations, compute distances, and even in understanding quadratic relationships.

When simplifying square roots, the goal is to express the number inside the square root as the product of a square number and another number. For example, to simplify \( \sqrt{12} \), recognize that 12 is \(4 \times 3\). Therefore:

\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

This process of simplification makes it easier to work with square roots, especially when dealing with larger numbers or performing algebraic operations.

The square root of 128 is represented as \( \sqrt{128} \). It is a value that, when multiplied by itself, gives the number 128. To understand it better, let's delve into its decimal and simplified radical forms.

In decimal form, the square root of 128 is approximately 11.3137. This means:

\[ \sqrt{128} \approx 11.3137 \]

However, to express the square root of 128 in its simplest radical form, we need to break it down into its prime factors. This process is known as simplifying the square root.

1. Find the prime factorization of 128. 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 into pairs:

   
   \[
   2^7 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2
   \]

3. Take one number out of each pair of identical numbers outside the square root:

   
   \[
   \sqrt{128} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2} = 2 \times 2 \times 2 \times \sqrt{2} = 8\sqrt{2}
   \]

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

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

This means that \( \sqrt{128} \) can be simplified to \( 8\sqrt{2} \), where 8 is the coefficient and \( \sqrt{2} \) is the simplified radical part.

Understanding the square root of 128 in both its decimal and simplified radical forms helps in various mathematical calculations and simplifies complex expressions involving radicals.

To simplify the square root of 128, we first need to find its prime factorization. Prime factorization involves breaking down a composite number into its prime factors, which are the prime numbers that multiply together to give the original number.

Here are the steps to find the prime factorization of 128:

1. Divide by the smallest prime number: Start by dividing 128 by the smallest prime number, which is 2.

   128 ÷ 2 = 64

2. Continue dividing by 2: Since 64 is still divisible by 2, continue the process.

   64 ÷ 2 = 32

3. Repeat the division:

   32 ÷ 2 = 16

4. Continue dividing:

   16 ÷ 2 = 8

5. Keep dividing:

   8 ÷ 2 = 4

6. One more time:

   4 ÷ 2 = 2

7. Finally:

   2 ÷ 2 = 1

Now, we can write 128 as a product of prime factors:

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Or, more concisely, using exponents:

128 = 2^7

This prime factorization will be useful for simplifying the square root of 128, as it allows us to group the prime factors into pairs, which can then be simplified further.

To simplify the square root of 128, we first need to break down the number 128 into its prime factors. Prime factorization is the process of expressing a number as a product of its prime numbers. Here’s a detailed step-by-step process to find the prime factors of 128:

1. Step 1: Start with the smallest prime number, which is 2. Check if 128 is divisible by 2.

• 128 ÷ 2 = 64

2. Step 2: Continue dividing by 2, as 64 is also divisible by 2.

• 64 ÷ 2 = 32

3. Step 3: Divide 32 by 2.

• 32 ÷ 2 = 16

4. Step 4: Divide 16 by 2.

• 16 ÷ 2 = 8

5. Step 5: Divide 8 by 2.

• 8 ÷ 2 = 4

6. Step 6: Divide 4 by 2.

• 4 ÷ 2 = 2

7. Step 7: Finally, divide 2 by 2.

• 2 ÷ 2 = 1

After completing these steps, we find that the prime factorization of 128 is:

\[128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]

Or, in exponential form:

\[128 = 2^7\]

This means that the prime factors of 128 are all 2, and 128 can be expressed as \(2^7\).

To simplify the square root of 128, it is essential to express 128 as a product of its prime factors. Here’s a step-by-step guide to break down 128 into its prime factors:

1. Step 1: Start by dividing 128 by the smallest prime number, which is 2. Since 128 is even, it is divisible by 2.

• 128 ÷ 2 = 64

2. Step 2: Continue dividing the result by 2. Divide 64 by 2.

• 64 ÷ 2 = 32

3. Step 3: Divide 32 by 2.

• 32 ÷ 2 = 16

4. Step 4: Divide 16 by 2.

• 16 ÷ 2 = 8

5. Step 5: Divide 8 by 2.

• 8 ÷ 2 = 4

6. Step 6: Divide 4 by 2.

• 4 ÷ 2 = 2

7. Step 7: Finally, divide 2 by 2.

• 2 ÷ 2 = 1

After all these steps, we have divided 128 completely into its prime factors, which are all 2s:

\[128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]

In exponential form, this is expressed as:

\[128 = 2^7\]

Therefore, the prime factorization of 128 is \(2^7\), indicating that 128 can be written as the product of seven 2s.

Simplifying the square root of a number involves breaking down the number into its prime factors and then simplifying using the properties of square roots. Here's a detailed, step-by-step guide to simplifying the square root of 128 using its prime factors:

1. Prime Factorization:

   First, perform the prime factorization of 128. We break it down as follows:

   • 128 ÷ 2 = 64
   • 64 ÷ 2 = 32
   • 32 ÷ 2 = 16
   • 16 ÷ 2 = 8
   • 8 ÷ 2 = 4
   • 4 ÷ 2 = 2
   • 2 ÷ 2 = 1

   So, the prime factorization of 128 is \( 2^7 \).

2. Grouping the Prime Factors:

   Next, we group the prime factors in pairs because we are dealing with square roots:

   \[ 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \]

   We can pair these as:

   \[ 128 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2 \]

   Each pair of twos can be simplified as a single 2 outside the square root:

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

3. Simplify the Expression:

   Take the square root of each pair of twos:

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

   Multiply the terms outside the square root:

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

4. Final Simplified Form:

   The square root of 128 simplified is:

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

By following these steps, you can simplify the square root of any number using its prime factors. This method ensures that you get the simplest form of the square root.

To simplify the square root of 128, follow these step-by-step instructions:

1. Identify the Prime Factors:

   Begin by finding the prime factors of 128:

   • 128 ÷ 2 = 64
   • 64 ÷ 2 = 32
   • 32 ÷ 2 = 16
   • 16 ÷ 2 = 8
   • 8 ÷ 2 = 4
   • 4 ÷ 2 = 2
   • 2 ÷ 2 = 1

   So, the prime factorization of 128 is \(2^7\).

2. Group the Prime Factors:

   Next, group the prime factors in pairs of two:

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

   These can be grouped as:

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

3. Simplify Each Pair:

   Take the square root of each pair of twos:

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

   Which simplifies to:

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

4. Multiply the Simplified Factors:

   Combine the simplified factors:

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

5. Final Simplified Form:

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

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

By following these steps, you can simplify the square root of 128 and understand the process of simplifying square roots using prime factorization.

To express 128 in exponential form, we need to break it down into its prime factors. Here's the step-by-step process:

1. First, identify the prime factors of 128. We start by dividing 128 by the smallest prime number, which is 2.
2. Keep dividing by 2 until you can no longer divide evenly:
   • 128 ÷ 2 = 64
   • 64 ÷ 2 = 32
   • 32 ÷ 2 = 16
   • 16 ÷ 2 = 8
   • 8 ÷ 2 = 4
   • 4 ÷ 2 = 2
   • 2 ÷ 2 = 1
3. So, 128 can be expressed as a product of prime factors: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\).
4. This is equivalent to \(2^7\).

Therefore, the exponential form of 128 is \(2^7\).

We can use this form to simplify the square root of 128 in the next steps. Understanding the exponential form is crucial because it allows us to apply the properties of exponents and radicals effectively.

In summary:

\[128 = 2^7\]

When dealing with square roots of exponential forms, we can utilize the properties of exponents and square roots to simplify the expressions. Here's a step-by-step guide:

1. Start with the exponential form of the number. For example, 128 can be written as \(2^7\).
2. Apply the property of square roots that states \(\sqrt{a^b} = a^{b/2}\). In our case, this becomes \(\sqrt{2^7} = 2^{7/2}\).
3. Simplify the exponent. Here, \(2^{7/2}\) can be split into two parts: \(2^3 \cdot 2^{1/2}\).
4. Recognize that \(2^3 = 8\) and \(2^{1/2} = \sqrt{2}\). Therefore, the expression simplifies to \(8\sqrt{2}\).

So, the square root of \(2^7\) or \(\sqrt{128}\) simplifies to \(8\sqrt{2}\). This method is powerful for simplifying square roots of numbers that can be expressed as exponential forms.

Here's the detailed breakdown in MathJax:

• \(\sqrt{128} = \sqrt{2^7}\)
• \(\sqrt{2^7} = 2^{7/2}\)
• \(2^{7/2} = 2^3 \cdot 2^{1/2}\)
• \(2^3 = 8\) and \(2^{1/2} = \sqrt{2}\)
• Therefore, \(\sqrt{128} = 8\sqrt{2}\)

After breaking down the number 128 into its prime factors and simplifying each part, the next step is to combine the results to find the simplified form of the square root of 128.

1. Recall the factorization of 128:

   • Prime factors: \( 128 = 2^7 \)

2. Rewrite the square root of 128 using its prime factors:

   • \( \sqrt{128} = \sqrt{2^7} \)

3. Express the exponent in terms of powers of 2:

   • \( \sqrt{2^7} = \sqrt{2^6 \cdot 2} \)

4. Separate the square root of the product into the product of square roots:

   • \( \sqrt{2^6 \cdot 2} = \sqrt{2^6} \cdot \sqrt{2} \)

5. Simplify the square root of \( 2^6 \):

   • \( \sqrt{2^6} = 2^3 = 8 \)

6. Combine the results to get the simplified form of the square root of 128:

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

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

To find the final simplified form of the square root of 128, we start by expressing 128 in terms of its prime factors:

128 can be written as:

\[ 128 = 2^7 \]

Next, we express the square root of 128 using its prime factorization:

\[ \sqrt{128} = \sqrt{2^7} \]

We can simplify this by taking out pairs of 2s from under the square root:

\[ \sqrt{2^7} = \sqrt{2^6 \times 2} = \sqrt{(2^3)^2 \times 2} \]

Since the square root of \( (2^3)^2 \) is \( 2^3 \), we get:

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

Simplifying further, we find:

\[ 2^3 \sqrt{2} = 8\sqrt{2} \]

Thus, the final simplified form of the square root of 128 is:

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

This means that the square root of 128, when simplified, is \( 8\sqrt{2} \).

In decimal form, this is approximately:

\[ 8 \times 1.414 = 11.313 \]

Therefore, \( \sqrt{128} \approx 11.313 \) in decimal form.

When simplifying the square root of 128, there are several common mistakes that students often make. Being aware of these can help you avoid them and ensure you simplify square roots correctly.

1. Incorrect Prime Factorization: One frequent mistake is not breaking down 128 correctly into its prime factors. Remember, the prime factorization of 128 is \( 2^7 \). Ensure you list all the factors accurately.

   Correct prime factorization of 128: \( 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \) or \( 2^7 \).

2. Ignoring Pairing of Factors: When taking the square root, it's essential to pair the prime factors. Each pair of \( 2 \) inside the radical sign (\( \sqrt{} \)) will come out as a single 2.

   For \( 128 = 2^7 \), grouping pairs: \( (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 2 \). Each pair of \( 2 \) simplifies to a single 2 outside the radical.

3. Forgetting to Simplify Completely: Sometimes, students stop too early and do not simplify the radical fully. Ensure you bring all possible factors out of the radical sign.

   Simplify \( \sqrt{128} = \sqrt{2^7} = 8\sqrt{2} \).

4. Misunderstanding Radical Notation: Confusing the radical notation can lead to mistakes. Remember that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

   For \( \sqrt{128} \), we use \( 128 = 64 \times 2 \), hence \( \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} \).

5. Neglecting to Check Work: Always recheck your steps to ensure no factor or step was missed. Verification helps catch and correct errors.

   After simplifying, re-calculate to confirm \( \sqrt{128} = 8\sqrt{2} \).

By being mindful of these common mistakes, you can improve your skills in simplifying square roots and avoid errors.

Here are some practice problems and examples to help you master the simplification of square roots using the square root of 128 as a guide:

1. Simplify the square root of 72:

   • Step 1: Find the prime factors of 72: \(72 = 2^3 \times 3^2\)
   • Step 2: Separate the factors under the square root: \(\sqrt{72} = \sqrt{2^3 \times 3^2}\)
   • Step 3: Simplify by taking the square roots of the factors: \(\sqrt{72} = \sqrt{2^2 \times 2 \times 3^2} = 6\sqrt{2}\)

2. Simplify the square root of 288:

   • Step 1: Find the prime factors of 288: \(288 = 2^5 \times 3^2\)
   • Step 2: Separate the factors under the square root: \(\sqrt{288} = \sqrt{2^5 \times 3^2}\)
   • Step 3: Simplify by taking the square roots of the factors: \(\sqrt{288} = \sqrt{2^4 \times 2 \times 3^2} = 12\sqrt{2}\)

3. Simplify the square root of 108:

   • Step 1: Find the prime factors of 108: \(108 = 2^2 \times 3^3\)
   • Step 2: Separate the factors under the square root: \(\sqrt{108} = \sqrt{2^2 \times 3^3}\)
   • Step 3: Simplify by taking the square roots of the factors: \(\sqrt{108} = 6\sqrt{3}\)

4. Simplify the square root of 50:

   • Step 1: Find the prime factors of 50: \(50 = 2 \times 5^2\)
   • Step 2: Separate the factors under the square root: \(\sqrt{50} = \sqrt{2 \times 5^2}\)
   • Step 3: Simplify by taking the square roots of the factors: \(\sqrt{50} = 5\sqrt{2}\)

5. Simplify the square root of 180:

   • Step 1: Find the prime factors of 180: \(180 = 2^2 \times 3^2 \times 5\)
   • Step 2: Separate the factors under the square root: \(\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5}\)
   • Step 3: Simplify by taking the square roots of the factors: \(\sqrt{180} = 6\sqrt{5}\)

These examples should give you a good understanding of the process. Practice with these problems and try creating your own for additional practice!

When it comes to simplifying square roots, understanding advanced techniques can make complex problems more manageable. Here are some methods to simplify square roots beyond the basics:

1. Product and Quotient Rules

The Product Rule and Quotient Rule for square roots allow us to break down and simplify more complicated expressions.

• Product Rule: If a and b are non-negative, then √(ab) = √a × √b.
• Quotient Rule: If a and b are non-negative and b ≠ 0, then √(a/b) = √a / √b.

2. Simplifying Radical Expressions

For more complex expressions, factor perfect squares from the radicand (the number inside the square root). This technique involves:

1. Factoring the radicand into its prime factors.
2. Grouping the prime factors into pairs.
3. Taking one number from each pair out of the square root.

For example, to simplify √300:

• Factor 300: 300 = 2 × 2 × 3 × 5 × 5.
• Group pairs: √(2 × 2 × 3 × 5 × 5) = √(2² × 3 × 5²).
• Simplify: 2 × 5 × √3 = 10√3.

3. Combining Radical Expressions

Combine multiple radical expressions into a single expression before simplifying:

For example, simplify √12 × √3:

• Combine: √(12 × 3) = √36.
• Simplify: √36 = 6.

4. Rationalizing the Denominator

When dealing with fractions, it is often useful to rationalize the denominator by removing the square root from the denominator:

For example, simplify 1 / √2:

• Multiply numerator and denominator by √2: 1 / √2 × √2 / √2 = √2 / 2.

5. Using Exponential Form

Expressing numbers in exponential form can simplify the process. For instance:

To simplify √128:

• Express 128 as a product of prime factors: 128 = 2⁷.
• Rewrite using exponent rules: √(2⁷) = 2^3.5.
• Separate into a product of square roots: 2³ × √2 = 8√2.

6. Simplifying Higher-Order Roots

When dealing with roots higher than the square root, apply similar rules. For example:

To simplify ∛(54):

• Factor 54 into primes: 54 = 2 × 3³.
• Separate: ∛(2 × 3³) = 3∛2.

Practice Problems

Practice these techniques to become proficient. Here are a few problems to try:

• Simplify √72.
• Simplify √50 / √2.
• Simplify ∛(64).

By mastering these advanced techniques, you can handle more complex radical expressions with confidence.

In this guide, we explored the process of simplifying the square root of 128. Simplifying square roots involves breaking down the number into its prime factors and then applying square root properties to simplify the expression.

Here's a quick summary of the steps we followed to simplify \( \sqrt{128} \):

1. First, we determined the prime factorization of 128, which is \( 128 = 2^7 \).
2. Next, we rewrote \( \sqrt{128} \) as \( \sqrt{2^7} \).
3. We used the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the factors, focusing on finding perfect squares within the prime factors.
4. Recognizing that \( 2^6 \) (or \( 64 \)) is a perfect square, we split the expression as \( \sqrt{2^6 \times 2} \).
5. We then simplified \( \sqrt{2^6} \) to \( 2^3 \) because \( \sqrt{(2^3)^2} = 2^3 = 8 \).
6. This allowed us to express \( \sqrt{128} \) as \( \sqrt{2^6 \times 2} = 8 \times \sqrt{2} \).

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

Here is a table summarizing the key steps:

By following these steps, we can simplify any square root, even those involving larger or more complex numbers. Remember to always look for perfect squares within the factorization to make the process easier.

Understanding these fundamental steps is crucial for simplifying square roots and can be applied to more advanced mathematical problems involving radicals.

For further practice, try simplifying other square roots or explore how these principles apply to other mathematical concepts. Check out the additional resources and references for more examples and practice problems.

To deepen your understanding of simplifying square roots, including the square root of 128, consider exploring the following resources and references. These materials provide comprehensive explanations, step-by-step guides, practice problems, and advanced techniques for working with radicals.

• Interactive Tutorials and Videos

  • This video tutorial covers the basics of simplifying square roots with clear examples and interactive exercises.

  • A step-by-step walkthrough of simplifying \( \sqrt{128} \), breaking down the process into easy-to-follow stages.

• Online Calculators and Tools

  • Use this tool to simplify square roots and verify your results quickly.

  • A versatile calculator that not only simplifies square roots but also provides step-by-step solutions.

• Practice Problems and Worksheets

  • Download worksheets to practice simplifying various square roots and enhance your skills.

  • Access a collection of problems specifically designed to practice simplifying radicals.

• Textbooks and Reference Materials

  • This free textbook covers a wide range of topics in algebra, including detailed sections on radicals and their properties.

  • An online resource offering lessons, practice, and quizzes on simplifying square roots.

• Advanced Topics

  • Explore more advanced concepts involving radicals and learn how to handle complex expressions and operations.

  • This course offers an in-depth look at radicals and their simplification, perfect for those looking to extend their understanding.

These resources will provide you with the knowledge and tools you need to master the art of simplifying square roots and enhance your overall mathematical skills.