Simplify Square Root 24 Easily: Step-by-Step Guide

The square root of 24 can be simplified using the prime factorization method. Here's a detailed step-by-step process:

Step-by-Step Simplification Process

1. Start with the number 24.
2. Find the prime factors of 24:
   • 24 is divisible by 2 (24 ÷ 2 = 12)
   • 12 is divisible by 2 (12 ÷ 2 = 6)
   • 6 is divisible by 2 (6 ÷ 2 = 3)
   • 3 is a prime number
3. So, the prime factorization of 24 is 2 × 2 × 2 × 3.
4. Rewrite 24 as a product of prime factors: \( 24 = 2^3 \times 3 \).
5. Group the factors in pairs of two: \( 24 = (2^2) \times 2 \times 3 \).
6. Simplify the square root using these pairs:

   \( \sqrt{24} = \sqrt{(2^2) \times 2 \times 3} \)

7. Since \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \):

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

8. Calculate the square root of the perfect square:

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

9. Combine the simplified parts:

   \( 2 \times \sqrt{6} \)

Conclusion

Therefore, the simplified form of \( \sqrt{24} \) is \( 2\sqrt{6} \).

Square roots are fundamental mathematical operations that are essential in various fields, including algebra, geometry, and calculus. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 24 is written as \( \sqrt{24} \).

To understand square roots better, let's break down the process of simplifying the square root of 24.

• Step 1: Prime Factorization

  
  The first step is to express 24 as a product of its prime factors: \( 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \).

• Step 2: Group the Factors

  
  Group the factors into pairs: \( 24 = (2 \times 2) \times 2 \times 3 = 4 \times 6 \).

• Step 3: Simplify the Square Root

  
  Rewrite the expression inside the square root as the product of square roots: \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \).

• Step 4: Calculate the Square Root

  
  Simplify further by calculating the square root of 4, which is 2: \( \sqrt{4} = 2 \). Thus, \( \sqrt{24} = 2\sqrt{6} \).

Therefore, the square root of 24 simplifies to \( 2\sqrt{6} \), which is approximately 4.899 in decimal form.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 24 is a number that, when squared, results in 24. The notation for square root is the radical symbol (√) followed by the number under the root, called the radicand.

To simplify the square root of 24, we can follow these steps:

1. Factor 24 into its prime factors: \(24 = 2^3 \times 3\).
2. Group the factors into pairs of identical numbers: \(24 = (2^2) \times 6\).
3. Take the square root of each pair and move it outside the radical: \(\sqrt{24} = \sqrt{(2^2) \times 6} = 2\sqrt{6}\).

The simplified form of the square root of 24 is \(2\sqrt{6}\). This form is more manageable for calculations and provides a clearer understanding of the number's structure.

In decimal form, the square root of 24 is approximately 4.89898. This approximation is useful for practical calculations where an exact radical form is not required.

To summarize:

• The square root of 24 is \(2\sqrt{6}\).
• In decimal form, it is approximately 4.89898.
• This simplification helps in understanding and calculating with square roots.

Square roots are fundamental in mathematics, often representing the inverse operation of squaring a number. Understanding the properties of square roots is essential for simplifying and manipulating radical expressions. Here are some basic properties of square roots:

• Definition: The square root of a number a is a value x such that \( x^2 = a \). It is denoted as \( \sqrt{a} \).
• Positive and Negative Roots: Every positive real number has two square roots: one positive and one negative. For example, \( \sqrt{24} = 2\sqrt{6} \) and \( -\sqrt{24} = -2\sqrt{6} \).
• Principal Square Root: The principal square root is the non-negative root of a number. When we refer to \( \sqrt{a} \), we usually mean the principal square root.
• Product Property: The square root of a product is equal to the product of the square roots of the factors: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \). For example, \( \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \).
• Quotient Property: The square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). For instance, \( \sqrt{\frac{24}{4}} = \frac{\sqrt{24}}{\sqrt{4}} = \frac{2\sqrt{6}}{2} = \sqrt{6} \).
• Irrational Numbers: The square root of a non-perfect square is an irrational number. For example, \( \sqrt{24} = 2\sqrt{6} \) is irrational because \( \sqrt{6} \) is irrational.
• Simplification: To simplify a square root, factor the number into its prime factors and pair the prime factors to bring them out of the square root. For example, \( \sqrt{24} = \sqrt{2^3 \cdot 3} = \sqrt{2^2 \cdot 6} = 2\sqrt{6} \).

Simplifying square roots involves reducing the expression under the radical to its simplest form. Here, we will explore various methods to achieve this, with a focus on simplifying the square root of 24. The most common methods are:

• Prime Factorization Method
• Long Division Method

Prime Factorization Method

This method involves expressing the number as a product of its prime factors and then simplifying.

1. List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
2. Identify the perfect squares: 1, 4.
3. Divide 24 by the largest perfect square: \( 24 \div 4 = 6 \).
4. Calculate the square root of the perfect square: \( \sqrt{4} = 2 \).
5. Combine the results: \( \sqrt{24} = 2 \sqrt{6} \).

Long Division Method

The long division method is used to find the square root of a number to a desired degree of accuracy. It is especially useful for non-perfect squares.

1. Write 24 as a decimal (24.000000).
2. Pair the digits from right to left (24).
3. Find the largest number whose square is less than or equal to 24 (4).
4. Subtract and bring down the next pair of zeros (24 - 16 = 8).
5. Double the quotient (4) and find a new digit to complete the divisor (8x2 = 16).
6. Continue the process to get more decimal places as needed (4.898979).

Exact Form and Decimal Form

The simplified form of \( \sqrt{24} \) can be represented as:

• Exact Form: \( 2 \sqrt{6} \)
• Decimal Form: 4.89898 (approx.)

Using these methods, you can simplify any square root, making it easier to work with in various mathematical problems.

The prime factorization method is an effective way to simplify square roots by breaking down the number into its prime factors. Here’s a step-by-step guide to simplifying the square root of 24 using prime factorization:

1. First, find the prime factors of 24. The prime factors of 24 are 2 and 3.
2. Express 24 as a product of these prime factors:

   \[ 24 = 2 \times 2 \times 2 \times 3 \]

3. Group the prime factors into pairs:

   \[ 24 = (2 \times 2) \times 2 \times 3 = 2^2 \times 2 \times 3 \]

4. Take the square root of each pair of prime factors:

   \[ \sqrt{24} = \sqrt{2^2 \times 2 \times 3} \]

5. Since the square root of \(2^2\) is 2, take this out of the square root:

   \[ \sqrt{24} = 2\sqrt{2 \times 3} \]

6. Multiply the remaining numbers inside the square root:

   \[ \sqrt{24} = 2\sqrt{6} \]

Therefore, the simplified form of \(\sqrt{24}\) is \(2\sqrt{6}\).

This method works well for any number that can be broken down into prime factors, making it a versatile tool for simplifying square roots.

Simplifying square roots of non-perfect squares involves breaking down the radicand (the number inside the square root) into its prime factors and using the product rule for radicals. Here is a detailed step-by-step method to simplify non-perfect square roots:

1. Factorize the Radicand:

   Identify the prime factors of the number inside the square root. For example, for √24, the prime factorization is:

   \[
   24 = 2^3 \times 3
   \]

2. Group the Factors:

   Pair the prime factors into perfect squares. Here, 24 can be grouped as:

   \[
   24 = (2^2) \times (2 \times 3)
   \]

3. Rewrite Using the Product Rule:

   Express the radicand as a product of square roots:

   \[
   \sqrt{24} = \sqrt{(2^2) \times (2 \times 3)} = \sqrt{2^2} \times \sqrt{2 \times 3}
   \]

4. Simplify the Square Roots:

   Simplify each square root. The square root of a perfect square (like 2²) is the base number (2), and the square root of a product remains under the root:

   \[
   \sqrt{24} = 2 \times \sqrt{6}
   \]

Let's see another example with √50:

• Factorize the Radicand:

  \[
  50 = 2 \times 5^2
  \]

• Rewrite Using the Product Rule:

  \[
  \sqrt{50} = \sqrt{2 \times 5^2} = \sqrt{2} \times \sqrt{5^2}
  \]

• Simplify the Square Roots:

  \[
  \sqrt{50} = \sqrt{2} \times 5 = 5\sqrt{2}
  \]

In general, the product rule for radicals states:

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

Using this rule, you can simplify any non-perfect square by identifying its prime factors, grouping them into perfect squares, and simplifying each square root separately.

Simplifying the square root of 24 involves a few steps to break it down into its simplest form. Follow these steps to understand the process:

1. Identify Factors: List the factors of 24, which are 1, 2, 3, 4, 6, 8, 12, and 24.
2. Find Perfect Squares: From the list of factors, identify the perfect squares. The perfect squares in this case are 1 and 4.
3. Divide by the Largest Perfect Square: Divide 24 by the largest perfect square found in the previous step.
   • 24 ÷ 4 = 6
4. Calculate the Square Root: Calculate the square root of the largest perfect square.
   • √4 = 2
5. Simplify: Combine the results of steps 3 and 4 to get the square root of 24 in its simplest form:
   • √24 = 2√6

The simplified form of √24 is therefore 2√6. This means that 24 can be expressed as the product of 2 and the square root of 6.

Using this method, you can simplify other non-perfect square roots by following a similar process of identifying factors, finding perfect squares, and combining the results.

In this section, we will look at some examples of how to simplify square roots step-by-step. This will help reinforce the concepts discussed earlier and provide practical illustrations of the simplification process.

Example 1: Simplify √24

1. Factor the number inside the square root:
   \( 24 = 2 \times 2 \times 2 \times 3 \)
2. Group the prime factors into pairs:
   \( 24 = (2 \times 2) \times 6 \)
3. Take the square root of each group:
   \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \)

Example 2: Simplify √45

1. Factor the number inside the square root:
   \( 45 = 3 \times 3 \times 5 \)
2. Group the prime factors into pairs:
   \( 45 = (3 \times 3) \times 5 \)
3. Take the square root of each group:
   \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \)

Example 3: Simplify √50

1. Factor the number inside the square root:
   \( 50 = 2 \times 5 \times 5 \)
2. Group the prime factors into pairs:
   \( 50 = 2 \times (5 \times 5) \)
3. Take the square root of each group:
   \( \sqrt{50} = \sqrt{2 \times 25} = \sqrt{2} \times \sqrt{25} = \sqrt{2} \times 5 = 5\sqrt{2} \)

Example 4: Simplify √18

1. Factor the number inside the square root:
   \( 18 = 2 \times 3 \times 3 \)
2. Group the prime factors into pairs:
   \( 18 = 2 \times (3 \times 3) \)
3. Take the square root of each group:
   \( \sqrt{18} = \sqrt{2 \times 9} = \sqrt{2} \times \sqrt{9} = \sqrt{2} \times 3 = 3\sqrt{2} \)

Example 5: Simplify √75

1. Factor the number inside the square root:
   \( 75 = 3 \times 5 \times 5 \)
2. Group the prime factors into pairs:
   \( 75 = 3 \times (5 \times 5) \)
3. Take the square root of each group:
   \( \sqrt{75} = \sqrt{3 \times 25} = \sqrt{3} \times \sqrt{25} = \sqrt{3} \times 5 = 5\sqrt{3} \)

Example 6: Simplify 2√12 + 9√3

1. Simplify \( 2\sqrt{12} \):
   \( 12 = 4 \times 3 \)
   \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
   Therefore, \( 2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3} \)
2. Combine the terms:
   \( 4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3} \)

The square root of 24 can be simplified through the following steps:

1. Prime Factorization Method:

   • First, factorize 24 into its prime factors:

   \( 24 = 2 \times 2 \times 2 \times 3 \)

   • Group the prime factors into pairs:

   \( 24 = (2 \times 2) \times 6 \)

   • Take the square root of each group of pairs and any remaining factor:

   \( \sqrt{24} = \sqrt{(2 \times 2) \times 6} = 2 \sqrt{6} \)

2. Simplification Process:

   • Rewrite 24 as a product of its prime factors:

   \( 24 = 2^2 \times 6 \)

   • Express the square root of the product as the product of square roots:

   \( \sqrt{24} = \sqrt{2^2 \times 6} \)

   • Separate the square root of the perfect square from the square root of the remaining factor:

   \( \sqrt{24} = \sqrt{2^2} \times \sqrt{6} \)

   • Simplify the square root of the perfect square:

   \( \sqrt{24} = 2 \sqrt{6} \)

Thus, the simplified form of the square root of 24 is \( 2 \sqrt{6} \).

In decimal form, this value is approximately 4.898979.

When simplifying square roots, several common mistakes can occur. Here are some important points to keep in mind to avoid errors:

1. Incorrect Prime Factorization:

   • Ensure that you break down the number into its correct prime factors. For example, \( 24 = 2 \times 2 \times 2 \times 3 \), not \( 24 = 4 \times 6 \).

2. Forgetting to Simplify Completely:

   • After factorizing, always check if the square root can be further simplified. For example, \(\sqrt{24}\) simplifies to \(2\sqrt{6}\), not just \(\sqrt{24}\).

3. Misplacing Factors Outside the Radical:

   • Only take out pairs of prime factors from under the square root. For instance, in \(\sqrt{24} = \sqrt{2^2 \times 6}\), only \(2\) is taken out to get \(2\sqrt{6}\).

4. Incorrectly Handling Non-Perfect Squares:

   • Non-perfect squares should remain under the square root. For example, in \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\), \(2\) stays inside the radical.

5. Arithmetic Errors:

   • Double-check your multiplication and division steps when dealing with the factors. Small errors can lead to incorrect simplification.

By being mindful of these common mistakes, you can ensure accurate and simplified results when working with square roots.

Simplifying square roots has various practical applications in real-world scenarios. Here are some examples:

1. Architecture and Construction

In architecture and construction, square roots are used to calculate the dimensions of different shapes. For instance, if a designer needs to create a square lawn with an area of 24 square meters, they would simplify the square root of 24 to find the side length.

• Calculate the area: \( \text{Area} = \text{side}^2 \)
• Find the side length: \( \text{side} = \sqrt{24} \approx 2\sqrt{6} \approx 4.9 \) meters

2. Physics and Engineering

Square roots play a critical role in physics and engineering. For example, the time it takes for an object to fall from a certain height can be determined using square roots. If an object is dropped from a height \( h \) feet, the time \( t \) it takes to reach the ground is given by:

• Formula: \( t = \frac{\sqrt{h}}{4} \)
• Example: Dropped from 64 feet, \( t = \frac{\sqrt{64}}{4} = \frac{8}{4} = 2 \) seconds

3. Accident Investigation

In accident investigations, police use square roots to determine the speed of a vehicle before braking. The length of skid marks \( d \) feet helps to estimate the speed \( v \) in miles per hour:

• Formula: \( v = \sqrt{24d} \)
• Example: For 190 feet skid marks, \( v = \sqrt{24 \times 190} \approx 67.5 \) mph

4. Medicine and Biology

In medicine, the dosage of certain medications may depend on the body surface area, which can be calculated using the square root of the height and weight measurements.

• Formula: \( BSA = \sqrt{\frac{\text{height (cm)} \times \text{weight (kg)}}{3600}} \)

5. Finance

Square roots are used in finance to calculate standard deviations, which measure the risk or volatility of investments. Simplifying square roots helps in making these calculations more manageable.

• Formula: \( \text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \)

Understanding and simplifying square roots is essential for solving practical problems efficiently and accurately across various fields.

Practice is key to mastering the simplification of square roots. Below are a series of practice problems to help you apply the methods of simplifying square roots. Try solving these problems step-by-step, and check your answers to ensure accuracy.

• Problem 1: Simplify \( \sqrt{18} \)

  1. Identify the prime factors of 18: \( 18 = 2 \times 3^2 \)
  2. Rewrite the square root using these factors: \( \sqrt{18} = \sqrt{2 \times 3^2} \)
  3. Extract the square root of the perfect square: \( \sqrt{2 \times 3^2} = 3\sqrt{2} \)

  Answer: \( 3\sqrt{2} \)

• Problem 2: Simplify \( \sqrt{50} \)

  1. Identify the prime factors of 50: \( 50 = 2 \times 5^2 \)
  2. Rewrite the square root using these factors: \( \sqrt{50} = \sqrt{2 \times 5^2} \)
  3. Extract the square root of the perfect square: \( \sqrt{2 \times 5^2} = 5\sqrt{2} \)

  Answer: \( 5\sqrt{2} \)

• Problem 3: Simplify \( \sqrt{32} \)

  1. Identify the prime factors of 32: \( 32 = 2^5 \)
  2. Rewrite the square root using these factors: \( \sqrt{32} = \sqrt{2^5} \)
  3. Group the factors into pairs: \( \sqrt{2^5} = \sqrt{(2^2)^2 \times 2} \)
  4. Extract the square root of the perfect square: \( \sqrt{(2^2)^2 \times 2} = 4\sqrt{2} \)

  Answer: \( 4\sqrt{2} \)

• Problem 4: Simplify \( \sqrt{72} \)

  1. Identify the prime factors of 72: \( 72 = 2^3 \times 3^2 \)
  2. Rewrite the square root using these factors: \( \sqrt{72} = \sqrt{2^3 \times 3^2} \)
  3. Group the factors into pairs: \( \sqrt{2^3 \times 3^2} = \sqrt{(2^2 \times 3^2) \times 2} \)
  4. Extract the square root of the perfect square: \( \sqrt{(2^2 \times 3^2) \times 2} = 6\sqrt{2} \)

  Answer: \( 6\sqrt{2} \)

• Problem 5: Simplify \( \sqrt{98} \)

  1. Identify the prime factors of 98: \( 98 = 2 \times 7^2 \)
  2. Rewrite the square root using these factors: \( \sqrt{98} = \sqrt{2 \times 7^2} \)
  3. Extract the square root of the perfect square: \( \sqrt{2 \times 7^2} = 7\sqrt{2} \)

  Answer: \( 7\sqrt{2} \)

By practicing these problems, you'll gain confidence in simplifying square roots and become more adept at recognizing patterns and perfect squares.

• What is the square root of 24?

  The square root of 24 is equal to \(2\sqrt{6}\) or approximately 4.899.

• What is the square root of 24 simplified?

  The square root of 24 in its simplest form is \(2\sqrt{6}\).

• Is 24 a perfect square?

  No, 24 is not a perfect square. The square root of 24 is not an integer.

• Is the square root of 24 irrational?

  Yes, \(\sqrt{24} = 4.89897948557\) is an irrational number because it is non-terminating and non-repeating.

• What is the square of 24?

  The square of 24 is \(24^2 = 576\).

• How do you simplify square roots when the radicals are alike?

  You can add or subtract the numbers in square roots only when the values under the radical sign are equal. For example, to simplify \( \sqrt{192} \), you break it down into its prime factors: \( \sqrt{192} = \sqrt{2 \times 96} = \sqrt{2 \times 2 \times 48} = 8\sqrt{3} \).

• What is the simplified form of the square root of 50?

  The simplified form of the square root of 50 is \(5\sqrt{2}\), which approximately equals 7.071.

• What is the simplified form of the square root of 72?

  The simplified form of the square root of 72 is \(6\sqrt{2}\), which approximately equals 8.485.