How to Simplify the Square Root of 68

To simplify the square root of 68, we follow a systematic process to express it in its simplest radical form.

Steps to Simplify √68

1. List Factors:

   Identify all factors of 68: 1, 2, 4, 17, 34, 68.

2. Find Perfect Squares:

   From the list of factors, identify the perfect squares: 1, 4.

3. Divide by the Largest Perfect Square:

   Divide 68 by the largest perfect square (4):

   \[ 68 \div 4 = 17 \]

4. Calculate the Square Root of the Perfect Square:

   Find the square root of 4:

   \[ \sqrt{4} = 2 \]

5. Express in Simplest Radical Form:

   Combine the results of the division and the square root calculation:

   \[ \sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \]

Additional Forms

The square root of 68 can also be expressed in other forms:

• Decimal Form: \( \sqrt{68} \approx 8.246 \)
• Exponent Form: \( 68^{\frac{1}{2}} = 2 \times 17^{\frac{1}{2}} \)

Summary

By following these steps, we find that the simplest radical form of the square root of 68 is:

\[ \sqrt{68} = 2\sqrt{17} \]

This method ensures that the square root is fully simplified and expressed in its most concise form.

Simplifying square roots is a fundamental concept in algebra that involves expressing a square root in its simplest form. This process is crucial for solving various mathematical problems efficiently. Let's explore the step-by-step method to simplify the square root of 68.

The main steps involved in simplifying square roots are:

1. Identify the Factors: Begin by listing all factors of the number under the square root. For 68, the factors are 1, 2, 4, 17, 34, and 68.
2. Find Perfect Squares: From the list of factors, identify the perfect squares. The perfect squares in this list are 1 and 4.
3. Divide by the Largest Perfect Square: Divide the original number by the largest perfect square identified. For 68, this would be:

   \[ 68 \div 4 = 17 \]

4. Square Root of the Perfect Square: Calculate the square root of the largest perfect square:

   \[ \sqrt{4} = 2 \]

5. Combine Results: Multiply the square root of the perfect square by the square root of the quotient:

   \[ \sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \]

By following these steps, we simplify the square root of 68 to \( 2\sqrt{17} \). This method ensures that the square root is expressed in its simplest and most efficient form, making it easier to work with in various mathematical contexts.

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. For example, the square root of 68 is written as \( \sqrt{68} \). Simplifying square roots involves expressing them in the simplest radical form.

To simplify \( \sqrt{68} \), we first need to factorize 68 into its prime factors:

• List the factors of 68: 1, 2, 4, 17, 34, 68
• Identify the perfect squares among these factors: 1, 4

Since 4 is the largest perfect square factor of 68, we can rewrite the square root of 68 as follows:

\[
\sqrt{68} = \sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17}
\]

This process shows that \( \sqrt{68} \) simplifies to \( 2\sqrt{17} \).

Understanding how to simplify square roots is crucial for solving more complex algebraic equations and for a deeper comprehension of number theory. By breaking down numbers into their prime factors and recognizing perfect squares, we can simplify any square root to its simplest form, enhancing our problem-solving skills and mathematical fluency.

To simplify the square root of 68, follow these detailed steps:

1. List the factors of 68:
   • 1, 2, 4, 17, 34, 68
2. Identify the perfect squares from the list of factors:
   • 1, 4
3. Divide 68 by the largest perfect square:
   • 68 ÷ 4 = 17
4. Calculate the square root of the largest perfect square:
   • √4 = 2
5. Combine the results:

   Put Steps 3 and 4 together to express the square root of 68 in its simplest radical form:

   \(\sqrt{68} = 2\sqrt{17}\)

6. Verify the result:

   The exact form of \(\sqrt{68}\) is \(2\sqrt{17}\), and the decimal form is approximately 8.246.

The prime factorization method is an effective way to simplify square roots. Let's apply this method to simplify the square root of 68 step by step:

1. First, find the prime factors of 68:
   • 68 can be factored into 2 × 34
   • 34 can be further factored into 2 × 17
   • Therefore, the prime factors of 68 are 2, 2, and 17
2. Group the prime factors in pairs:
   • The factors of 68 are \( 2^2 \times 17 \)
3. Take the square root of each pair:
   • \( \sqrt{2^2} = 2 \)
4. Multiply the results:
   • \( 2 \times \sqrt{17} \)

Thus, the simplified form of the square root of 68 is \( 2\sqrt{17} \). This method is useful for breaking down more complex radicals into simpler forms.

Simplifying square roots using perfect squares is a straightforward method. By identifying the perfect square factors of a number, we can simplify the expression significantly. Here’s a step-by-step process to simplify the square root of 68 using perfect squares:

1. Identify the factors of 68:
   • 1, 2, 4, 17, 34, 68
2. Find the perfect squares among the factors:
   • 1, 4
3. Choose the largest perfect square factor:
   • 4
4. Divide 68 by the largest perfect square factor:
   • \( \frac{68}{4} = 17 \)
5. Rewrite the square root expression:
   • \( \sqrt{68} = \sqrt{4 \times 17} \)
6. Separate the square root into two parts:
   • \( \sqrt{4} \times \sqrt{17} \)
7. Simplify the square root of the perfect square:
   • \( \sqrt{4} = 2 \)
8. Combine the results to get the simplest form:
   • \( 2\sqrt{17} \)

By following these steps, you can simplify \( \sqrt{68} \) to \( 2\sqrt{17} \). This method makes it easier to work with square roots in various mathematical problems.

To express the square root of 68 in its simplest radical form, we need to follow a series of steps. Here is the detailed process:

1. Identify Prime Factors:

   First, find the prime factors of 68. The prime factorization of 68 is \( 68 = 2^2 \times 17 \).

2. Rewrite the Square Root:

   Next, express the square root of 68 using these prime factors:

   \(\sqrt{68} = \sqrt{2^2 \times 17}\)

3. Simplify Using Perfect Squares:

   We can simplify the square root by taking the square root of the perfect square factor out of the radical:

   \(\sqrt{2^2 \times 17} = \sqrt{2^2} \times \sqrt{17} = 2\sqrt{17}\)

4. Final Expression:

   Thus, the square root of 68 in its simplest radical form is:

   \(\sqrt{68} = 2\sqrt{17}\)

This process helps in breaking down the square root into a more manageable and simplified form, which is especially useful in various mathematical applications.

The square root of 68 can be expressed in various forms, including its decimal and exponent representations. Below is a detailed explanation of both forms:

Decimal Form

To find the decimal form of √68, you can use a calculator to get a precise value:

\[\sqrt{68} \approx 8.246211251\]

This approximation is rounded to the nearest billionth. For most practical purposes, you can round it to the nearest hundredth:

\[\sqrt{68} \approx 8.25\]

Exponent Form

The exponent form of the square root is a convenient way to express the root using exponents. The square root of any number \(x\) can be written as:

\[x^{1/2}\]

Applying this to 68, we get:

\[68^{1/2}\]

Since we have simplified √68 to 2√17, we can also express it using exponents:

\[2 \times 17^{1/2}\]

Summary

• Decimal Form: \[\sqrt{68} \approx 8.246211251\]
• Exponent Form: \[68^{1/2} \text{ or } 2 \times 17^{1/2}\]

Practicing the simplification of square roots is essential for mastering the concept. Here are some detailed examples and practice problems to help you understand how to simplify √68:

Example 1: Simplifying √68

To simplify the square root of 68, follow these steps:

1. Prime Factorization: Break down 68 into its prime factors:

   \[68 = 2 \times 34 = 2 \times 2 \times 17 = 2^2 \times 17\]

2. Group the prime factors into pairs:

   \[2^2 \times 17\]

3. Take the square root of each pair and move them outside the radical:

   \[\sqrt{2^2 \times 17} = 2\sqrt{17}\]

So, the simplified form of √68 is \(2\sqrt{17}\).

Example 2: Simplifying Another Square Root

Let's simplify √72 using the same steps:

1. Prime Factorization:

   \[72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2\]

2. Group the prime factors into pairs:

   \[2^2 \times 2 \times 3^2\]

3. Take the square root of each pair and move them outside the radical:

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

So, the simplified form of √72 is \(6\sqrt{2}\).

Practice Problems

Try simplifying the following square roots on your own. Check your answers using the steps outlined above:

• Simplify \( \sqrt{50} \)
• Simplify \( \sqrt{98} \)
• Simplify \( \sqrt{45} \)
• Simplify \( \sqrt{80} \)

Answers:

• \( \sqrt{50} = 5\sqrt{2} \)
• \( \sqrt{98} = 7\sqrt{2} \)
• \( \sqrt{45} = 3\sqrt{5} \)
• \( \sqrt{80} = 4\sqrt{5} \)

When simplifying the square root of 68, it's crucial to avoid common mistakes to ensure accurate results. Here are some frequent errors and how to steer clear of them:

• Ignoring Perfect Squares: Always check for perfect squares within the radicand. For instance, 68 can be factored into 4 and 17, where 4 is a perfect square. Simplifying √68 correctly involves recognizing that √68 = √(4 × 17) = 2√17.
• Incomplete Factorization: Ensure you factor the number completely. Missing a factor can lead to incorrect simplification. For example, if you overlook the factorization of 68 as 4 × 17, you might miss the correct simplification step.
• Incorrect Application of the Product Rule: Misapplying the product rule can lead to errors. Remember that √ab = √a × √b only if a and b are correctly identified factors. Always recheck your factorization to apply this rule correctly.
• Skipping Simplification Steps: Simplification requires careful step-by-step processing. Rushing through steps can cause errors. Take your time to factorize, apply the square root rules, and simplify step by step.
• Forgetting to Combine Like Terms: When dealing with multiple radicals, ensure you combine like terms properly. For example, if simplifying multiple square roots, combine them appropriately to avoid errors.
• Decimal vs. Radical Form: Ensure you understand the difference between expressing results in decimal form and radical form. Simplified square roots often remain in radical form for exact values, e.g., 2√17 rather than converting to an approximate decimal.

By paying attention to these common pitfalls, you can simplify square roots accurately and efficiently. Practice regularly and double-check your work to build confidence and accuracy in your mathematical skills.

Below are some frequently asked questions about simplifying square roots:

• Q: How do you simplify a square root?

  A: To simplify a square root, factor the number under the radical into its prime factors and then pair the prime factors. For each pair, take one number out of the square root. For example, to simplify \( \sqrt{68} \):

  • Find the prime factorization of 68: \( 68 = 2 \times 34 = 2 \times 2 \times 17 \).
  • Pair the twos: \( \sqrt{68} = \sqrt{2^2 \times 17} \).
  • Take the 2 out of the radical: \( \sqrt{68} = 2\sqrt{17} \).
• Q: Can you add or subtract square roots?

  A: You can only add or subtract square roots that have the same radicand (the number under the radical). For example, \( 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3} \), but \( 2\sqrt{3} + 3\sqrt{2} \) cannot be simplified further.

• Q: What is the square root of a fraction?

  A: To find the square root of a fraction, take the square root of the numerator and the denominator separately. For example, \( \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \).

• Q: How do you simplify a square root of a variable expression?

  A: Simplify each part of the expression separately. For example, to simplify \( \sqrt{72z^{12}/2z^{10}} \):

  • First, simplify the fraction: \( \frac{72z^{12}}{2z^{10}} = 36z^2 \).
  • Then take the square root of each part: \( \sqrt{36z^2} = 6z \).
• Q: What is the simplified form of the square root of 50?

  A: To simplify \( \sqrt{50} \), factor 50 into its prime factors: \( 50 = 2 \times 25 = 2 \times 5^2 \). So, \( \sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2} \).

• Q: How do you simplify the square root of 72?

  A: First, factor 72: \( 72 = 2^3 \times 3^2 \). Then, simplify: \( \sqrt{72} = \sqrt{2^3 \times 3^2} = 3\sqrt{8} = 6\sqrt{2} \).

For those looking to deepen their understanding of simplifying square roots, there are a variety of resources and tools available online. Here are some valuable ones:

• Khan Academy: Offers comprehensive tutorials and practice exercises on simplifying square roots, including detailed video explanations. Visit their section.
• Math is Fun: Provides clear and easy-to-follow explanations on the process of simplifying square roots, complete with examples and interactive exercises. Check out their guide on .
• Symbolab Calculator: An advanced tool that not only simplifies square roots but also provides step-by-step solutions. It's an excellent resource for verifying your work. Try the .
• Paul's Online Math Notes: Offers detailed notes and practice problems on simplifying square roots, perfect for self-study. Visit the page.
• Wolfram Alpha: A powerful computational engine that can handle complex mathematical problems, including simplifying square roots. Use the tool.

These resources provide a range of methods and tools to help you master the simplification of square roots, whether you're just starting out or looking to refine your skills.